course Mth 151 here are the chapter one and 3.1 assignments. also i cannot find the access page to check your comments and i cannot find where my first test grade. please let me know how to access this information through my student email.
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20:30:47 `q001. Note that there are 6 questions in this assignment. Find the likely next element of the sequence 1, 2, 4, 7, 11, ... .
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RESPONSE --> 16 increases by 5
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20:30:53 The difference between 1 and 2 is 1; between 2 and 4 is 2; between 4 and 7 is 3; between 7 and 11 is 4. So we expect that the next difference will be 5, which will make the next element 11 + 5 = 16.
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RESPONSE --> ok
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20:33:01 `q002. Find the likely next two elements of the sequence 1, 2, 4, 8, 15, 26, ... .
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RESPONSE --> 42 64
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20:35:07 The difference between 1 and 2 is 1; the difference between 2 and 4 is 2, the difference between 4 and 8 is 4; the difference between 8 and 15 is 7; the difference between 15 and 26 is 11. The differences form the sequence 1, 2, 4, 7, 11, ... . As seen in the preceding problem the differences of this sequence are 1, 2, 3, 4, ... . We would expect the next two differences of this last sequence to be 5 and 6, which would extend the sequence 1, 2, 4, 7, 11, ... to 1, 2, 4, 7, 11, 16, 22, ... . If this is the continuation of the sequence of differences for the original sequence 1, 2, 4, 8, 15, 26, ... then the next two differences of this sequence would be 16 , giving us 26 + 16 = 42 as the next element, and 22, giving us 42 + 26 = 68 as the next element. So the original sequence would continue as 1, 2, 4, 8, 15, 26, 42, 68, ... .
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RESPONSE --> i think the 26 should be replaced with the 22 since its the next element giving 22 + 42=64
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20:35:47 `q003. What would be the likely next element in the sequence 1, 2, 4, 8, ... . It is understood that while this sequence starts off the same as that in the preceding exercise, it is not the same. The next element is not 15, and the pattern of the sequence is different than the pattern of the preceding.
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RESPONSE --> 16
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20:35:52 One obvious pattern for this sequence is that each number is doubled to get the next. If this pattern continues then the sequence would continue by doubling 8 to get 16. The sequence would therefore be 1, 2, 4, 8, 16, ... .
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RESPONSE --> ok
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20:38:03 `q004. There are two important types of patterns for sequences, one being the pattern defined by the differences between the numbers of the sequence, the other being the pattern defined by the ratios of the numbers of the sequence. In the preceding sequence 1, 2, 4, 8, 16, ..., the ratios were 2/1 = 2; 4/2 = 2; 8/4 = 2; 16/8 = 2. The sequence of ratios for 1, 2, 4, 8, 16, ..., is thus 2, 2, 2, 2, a constant sequence. Find the sequence of ratios for the sequence 32, 48, 72, 108, ... , and use your result to estimate the next number and sequence.
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RESPONSE --> 1.5 162
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20:38:08 The ratios are 48/32 = 1.5; 72 / 48 = 1.5; 108/72 = 1.5, so the sequence of ratios is 1.5, 1.5, 1.5, 1.5, ... . The next number the sequence should probably therefore be 108 * 1.5 = 162.
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RESPONSE --> ok
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20:42:17 `q005. Find the sequence of ratios for the sequence 1, 2, 3, 5, 8, 13, 21... , and estimate the next element of the sequence.
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RESPONSE --> 2, 1.5,1.6repeating, 1.6,1.625, 1.615
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20:42:35 The ratios are 2/1 = 2; 3/2 = 1.5; 5/3 = 1.66...; 8/5 = 1.60; 13/8 = 1.625; 21/13 = 1.615. The sequence of ratios is 2, 1.5, 1.66..., 1.625, 1.615, ... . We see that each number in the sequence lies between the two numbers that precede it -- 1.66... lies between 2 and 1.5; 1.60 lies between 1.5 and 1.66...; 1.625 lies between 1.66... and 1.60; 1.615 lies between 1.60 and 1.625. We also see that the numbers in the sequence alternate between being greater than the preceding number and less than the preceding number, so that the intervals between the numbers get smaller and smaller. So we expect that the next number in the sequence of ratios will be between 1.615 and 1.625, and if we pay careful attention to the pattern we expect the next number to be closer to 1.615 than to 1.625. We might therefore estimate that the next ratio would be about 1.618. We would therefore get 1.618 * 21 = 33.98 for the next number in the original sequence. However, since the numbers in the sequence are all whole numbers, we round our estimate up to 34. Our conjecture is that the sequence continues with 1, 2, 3, 5, 8, 13, 21, 34, ... .
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RESPONSE --> ok
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20:44:56 `q006. Without using ratios, can you find a pattern to the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, ..., and continue the sequence for three more numbers?
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RESPONSE --> 55, 89
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20:45:03 The pattern is that each element from the third on is the sum of the two elements that precede it. That is, 1+1=2, 2+1=3; 3+2=5; 5+3=8; 8+5=13; 13+8=21; 21+13=34; . The next three elements would therefore e 34+21=55; 55+34=89; 89+55=144. . The sequence is seen to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... .
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RESPONSE --> ok
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ðS}CwJβ Student Name: assignment #007
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20:49:02 `q001. Note that there are 7 questions in this assignment. Sketch three points A, B and C forming an equilateral triangle on a piece of paper, with point A at the lower left-hand corner, point B at the lower right-hand corner and point C at the top. Sketch the segments AB and AC. Now double the lengths of AB and AC, and place a point at each of the endpoints of these segments. Connect these new endpoints to form a new equilateral triangle. Two sides of this triangle will have three points marked while the new side will only have its two endpoints marked. Fix that by marking that middle point, so all three sides of your new triangle are marked the same. How many marked points were there in the original triangle, and how many are there in the new triangle?
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RESPONSE --> 3, 5
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20:49:25 The original triangle had the three points A, B and C. When you extended the two sides you marked the new endpoints, then you marked the point in the middle of the third side. So you've got 6 points marked. Click on 'Next Picture' to see the construction. The original points A, B and C are shown in red. The line segments from A to B and from A to C have been extended in green and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and an equally spaced point has been constructed at the midpoint of that side. Your figure should contain the three original points, plus the three points added when the new side was completed.
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RESPONSE --> ok
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20:51:15 `q002. Extend the two sides that meet at A by distances equal to the distance original lengths AC and AB and mark the endpoints of the newly extended segments. Each of the newly extended sides will have 4 marked points. Now connect the new endpoints to form a new right triangle. Mark points along the new side at the same intervals that occur on the other two sides. How many marked points are on your new triangle, and how many in the whole figure?
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RESPONSE --> 3, 9
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20:52:03 `q003. Continue the process for another step-extend each side by a distance equal to the original point-to-point distance. How many points do you have in the new triangle?
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RESPONSE --> 15
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20:52:13 You will add an endpoint to each newly extended side, so each of the new sides will contain 5 points. You will then have to add 3 equally spaced points to the new side, giving you a total of 13 points on the new triangle. In addition there are two marked points inside the triangle, for a total of 15 points. Click on 'Next Picture' to see the construction. The line segments along two sides of the triangle have again been extended and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and equally spaced points have been constructed along that side.
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RESPONSE --> ok
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20:52:19 `q004. Continue the process for one more step. How many points do you have in the new triangle?
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RESPONSE --> 21
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20:52:23 You will add an endpoint to each newly extended side, so each of the new sides will contain 6 points. You will then have to add 4 equally spaced points to the new side, giving you a total of 15 points on the new triangle. There are also 5 marked points inside the triangle for a total of 21 marked points.
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RESPONSE --> ok
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20:52:43 `q005. The sequence of marked points is 3, 6, 10, 15, 21. What do expect will be the next number in this sequence?
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RESPONSE --> 28
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20:52:47 The differences between these numbers are 3, 4, 5, 6. The next difference, according to this pattern, should be 7, which would make the next number 28.
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RESPONSE --> ok
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20:53:13 `q006. How can you tell, in terms of the process you used to construct these triangles, that the next number should be 7 greater?
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RESPONSE --> it was going up by +1 each time
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20:53:34 When you extend the triangle again, you will add two new endpoints and each side will now have 7 points. The 7 points on the new triangle will be all of the new points.
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RESPONSE --> ok
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20:54:36 `q007. How do you know this sequence will continue in this manner?
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RESPONSE --> n(n+1)/2
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20:54:40 Each time you extend the triangle, each side increases by 1. All the new marked points are on the new side, so the total number of marked points will increase by 1 more than with the previous extension.
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RESPONSE --> ok
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̃zαS Student Name: assignment #008
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20:57:43 `q001. There are seven questions in this set. See if you can figure out a strategy for quickly adding the numbers 1 + 2 + 3 + ... + 100, and give your result if you are successful. Don't spend more than a few minutes on your attempt.
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RESPONSE --> 101x50 =5050
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20:57:47 These numbers can be paired as follows: 1 with 100, 2 with 99, 3 with 98, etc.. There are 100 number so there are clearly 50 pairs. Each pair adds up to the same thing, 101. So there are 50 pairs each adding up to 101. The resulting sum is therefore total = 50 * 101 = 5050.
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RESPONSE --> ok
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20:59:26 `q002. See if you can use a similar strategy to add up the numbers 1 + 2 + ... + 2000.
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RESPONSE --> 2001x1000=2001000
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20:59:32 Pairing 1 with 2000, 2 with 1999, 3 with 1998, etc., and noting that there are 2000 numbers we see that there are 1000 pairs each adding up to 2001. So the sum is 1000 * 2001 = 2,001,000.
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RESPONSE --> ok
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21:00:21 `q003. See if you can devise a strategy to add up the numbers 1 + 2 + ... + 501.
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RESPONSE --> 502x26=13052
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21:01:17 We can pair 1 with 501, 2 with 500, 3 with 499, etc., and each pair will have up to 502. However there are 501 numbers, so not all of the numbers can be paired. The number in the 'middle' will be left out. However it is easy enough to figure out what that number is, since it has to be halfway between 1 and 501. The number must be the average of 1 and 501, or (1 + 501) / 2 = 502 / 2 = 266. Since the other 500 numbers are all paired, we have 250 pairs each adding up to 502, plus 266 left over in the middle. The sum is 250 * 502 + 266 = 125,500 + 266 = 125,751. Note that the 266 is half of 502, so it's half of a pair, and that we could therefore say that we effectively have 250 pairs and 1/2 pair, or 250.5 pairs. 250.5 is half of 501, so we can still calculate the number of pairs by dividing the total number of number, 501, by 2. The total sum is then found by multiplying this number of pairs by the sum 502 of each pair: 250.5 * 502 = 125,766.
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RESPONSE --> ok
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21:03:30 `q004. Use this strategy to add the numbers 1 + 2 + ... + 1533.
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RESPONSE --> 1534x750+767=1151267
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21:03:59 Pairing the numbers, 1 with 1533, 2 with 1532, etc., we get pairs which each adult to 1534. There are 1533 numbers so there are 1533 / 2 = 766.5 pairs. We thus have a total of 1534 * 766.5, whatever that multiplies out to (you've got a calculator, and I've only got my unreliable head).
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RESPONSE --> ok
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21:05:30 `q005. Use a similar strategy to add the numbers 55 + 56 + ... + 945.
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RESPONSE --> 16875
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21:06:23 We can pair up 55 and 945, 56 and 944, etc., obtaining 1000 for each pair. There are 945 - 55 + 1 = 891 numbers in the sum (we have to add 1 because 945 - 55 = 890 tells us how many 1-unit 'jumps' there are between 55 and 945--from 55 to 56, from 56 to 57, etc.. The first 'jump' ends up at 56 and the last 'jump' ends up at 945, so every number except 55 is the end of one of the 890 'jumps'. But 55 is included in the numbers to be summed, so we have 890 + 1 = 891 numbers in the sum). If we have 891 numbers in the sum, we have 891/2 = 445.5 pairs, each adding up to 1000. So we have a total of 445.5 * 1000 = 445,500.
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RESPONSE --> ok
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21:08:05 `q006. Devise a strategy to add the numbers 4 + 8 + 12 + 16 + ... + 900.
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RESPONSE --> 405454
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21:09:20 Pairing 4 with 900, 8 with 896, etc., we get pairs adding up to 904. The difference between 4 and 900 is 896. The numbers 'jump' by 4, so there are 896 / 4 = 224 'jumps'. None of these 'jumps' ends at the first number so there are 224 + 1 = 225 numbers. Thus we have 225 / 2 = 112.5 pairs each adding up to 904, and our total is 112.5 * 904.
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RESPONSE --> ok
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21:12:46 `q007. What expression would stand for the sum 1 + 2 + 3 + ... + n, where n is some whole number?
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RESPONSE --> find the sum of the whole numbers 1-n
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21:12:59 We can pair 1 and n, 2 and n-1, 3 and n-2, etc., in each case obtaining a sum of n + 1. There are n numbers so there are n/2 pairs, each totaling n + 1. Thus the total is n/2 * (n+1).
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RESPONSE --> ok
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LJٟ͙꜠z Student Name: assignment #011
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21:15:17 `q001. . There are five questions in this set. Draw four points in a square pattern (i.e., if the points were properly connected, they should form a square). From each of the points, draw a straight line to each of the other points. How many lines did you draw?
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RESPONSE --> 6
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21:15:34 Each corner of the square will connected to each of the other three corners, so from each corner you would have drawn three lines. Since there are four corners, had you followed the instructions precisely you would have drawn 4 * 3 = 12 lines. However each of these lines will be identical with another line you would have drawn, since for any two corners you would be drawing a line from the first to the second then another overlapping line from the second to the first. Therefore you might have said that there are 6 lines.
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RESPONSE --> ok
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21:16:11 `q002. How many triangles are there in the figure you drew?
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RESPONSE --> 4
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21:17:07 You should have a total of 8 triangles. The diagonals divide the square up into 4 small triangles. Each diagonal also divides the square into 2 larger triangles. Since there are 2 diagonals there are 4 larger triangles. The 4 small triangles and the 4 larger triangles total 8 triangles.
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RESPONSE --> ok
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21:20:47 `q003. The lines you drew should form a square with its two diagonals. Label the corners of the square A, B, C and D, going in order around the square, and label the center where the diagonals cross E. Now list all possible combinations of 3 of the letters A, B, C, D, E (note: combinations don't care about order, so A D E is the same as D A E or E A D or any other combination of these same three letters, so list each possible combination only once. That is, if you list for example ADE you won't list DAE).
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RESPONSE --> 30
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21:21:10 The easiest way to list these sequences is alphabetically: ABC, ABD, ABE all start with AB; then ACD and ACE start with AC and ADE starts with AD. This is a list of all possible combinations containing A. We next list all possible remaining combinations containing B: BCD, BCE and BDE. Then we write down CDE, the only remaining combination containing C. We thus have the 10 combinations ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE.
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RESPONSE --> ok
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21:22:54 `q004. Of the 10 combinations ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE, which form triangles on your figure?
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RESPONSE --> aec abe acb ced deb adb cbd acd
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21:23:14 ABC forms a large triangle from A to B to C and back to A. The same is true of ABD, ACD and BCD. These are the large triangles in the figure. ACE and BDE form straight lines, not triangles. ABE, ADE, BCE and CDE form small triangles. Thus of the 10 possible combinations of labeled points, we find the 4 large triangles and the 4 small triangles we saw earlier, in addition to 2 straight lines which do not form triangles. Since any triangle in the figure must be labeled by three of the five points A, B, C, D, E, we see that these are the only triangles that can be formed.
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RESPONSE --> ok
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21:24:01 `q005. Sketch the same figure as before, but without the line segment from A to B. Now how may triangles are there?
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RESPONSE --> 4
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21:24:06 Any possible triangle must still come from the list ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. We again see that ACE and BDE form straight lines so do not count as triangles. Now ABC, ABD and ABE do not form triangles because the line segment AB is now missing. This leaves us the five triangles ACD, ADE, BCE, BCE and CDE.
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RESPONSE --> ok
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zQTNŴ]ѫK{˹R Student Name: assignment #012
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21:26:16 `q001. Note that there are 4 questions in this assignment. Suppose I tell you 'If it rains today, I'll give you $100.' Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
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RESPONSE --> 2
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21:26:21 I said what would happen under a certain condition. In situation #2, that condition is fulfilled and what I said would happen doesn't happen. Therefore in situation #2 it is clear that I wasn't telling the truth. In situation #3, the condition that I addressed isn't fulfilled so no matter what happens I can't be accused of not telling the truth. I said what would happen if rains. No matter what happens, if it doesn't rain what I said cannot be held against me. It should be clear to anybody that situation #1 is exactly what you would expect, and that situation #4 is just would you would probably expect from my statement in the event that it doesn't rain, so nobody would say that this situation violates my claim to truthfulness.
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RESPONSE --> ok
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21:26:44 `q002. Suppose that tell you 'It will rain today and I will give you $100'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
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RESPONSE --> 2
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Je솢h雋}xazC Student Name: assignment #012
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21:26:56 It should be clear that situation #1 completely fulfills the conditions of my statement. Both of the things that I say will happen do happen. In situation #2, it rains but you don't get the $100. I said two things were going to happen and one of them didn't. In that case you would have to say that I wasn't telling truth. In situation #3, again one of the things I say is going to happen does but the other doesn't, so again you would have to say that I wasn't telling truth. In situation #4, neither of the things I say will happen does and certainly it would have to be said that I wasn't telling truth.
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RESPONSE --> ok
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21:28:34 `q003. Suppose that tell you 'It will rain today or I will give you $100, but not both'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
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RESPONSE --> 4
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21:28:51 In situations 2 and 3, one of the things happens and the other doesn't, so you would not be able to say that I wasn't telling the truth. However in situation 1, both things happen, which I said wouldn't be the case; and in situation 4 neither thing happens. In both of these situations you would have to say that I was not telling truth.
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RESPONSE --> ok
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21:30:30 `q004. Suppose that tell you 'It will rain today or I will give you $100'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
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RESPONSE --> 4
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21:31:08 At first this might seem to be the same as the preceding problem. But in the preceding problem we specifically said '... but not both.' In this case that qualification was not made. Therefore we have regard the statement as true as long as at least one of the conditions is fulfilled. This is certainly the case for situation 1: both conditions are true we can certainly say that at least one is true. So in situation #1 we have to regard the present statement as true. So situation #1 would not be included among those in which I could be accused of not telling the truth.
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RESPONSE --> ok
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w֓q~~weFF assignment #006 wO}nShwzmнK̹Ѓ Liberal Arts Mathematics I 10-03-2006
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19:26:56 Query 1.1.4 first 3 children male; conclusion next male. Inductive or deductive?
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RESPONSE --> deductive
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19:27:08 ** The argument is inductive, because it attempts to argue from a pattern. **
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RESPONSE --> ok
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19:27:46 Query 1.1.8 all men mortal, Socrates a man, therefore Socrates mortal.
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RESPONSE --> deductive. facts that apply to this case
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19:28:10 Query 1.1.20 1 / 3, 3 / 5, 5/7, ... Probable next element.
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RESPONSE --> ok
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19:29:20 **The numbers 1, 3, 5, 7 are odd numbers. We note that the numerators consist of the odd numbers, each in its turn. The denominator for any given fraction is the next odd number after the numerator. Since the last member listed is 5/7, with numerator 5, the next member will have numerator 7; its denominator will be the next odd number 9, and the fraction will be 7/9. There are other ways of seeing the pattern. We could see that we use every odd number in its turn, and that the numerator of one member is the denominator of the preceding member. Alternatively we might simply note that the numerator and denominator of the next member are always 2 greater than the numerator and denominator of the present member. **
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RESPONSE --> think i skipped forward on accident
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19:33:50 Query 1.1.23 1, 8, 27, 64, ... Probable next element.
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RESPONSE --> 125
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19:34:02 ** This is the sequence of cubes. 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125. The next element is 6^3 = 216. Successive differences also work: 1 8 27 64 125 .. 216 7 19 37 61 .. 91 12 18 24 .. 30 6 6 .. 6 **
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RESPONSE --> ok
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19:35:19 Query 1.1.36 11 * 11 = 121, 111 * 111 = 12321 1111 * 1111 = 1234321; next equation, verify.
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RESPONSE --> 11111x 11111= 12345432e+08
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19:35:26 ** We easily verify that 11111*11111=123,454,321 **
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RESPONSE --> ok
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19:38:06 Do you think this sequence would continue in this manner forever? Why or why not?
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RESPONSE --> yes, because of the sequence
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19:38:37 ** You could think forward to the next few products: What happens after you get 12345678987654321? Is there any reason to expect that the sequence could continue in the same manner? The middle three digits in this example are 8, 9 and 8. The logical next step would have 9, 10, 9, but now you would have 9109 in the middle and the symmetry of the number would be destroyed. There is every reason to expect that the pattern would also be destroyed. **
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RESPONSE --> ok
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19:39:06 Query 1.1.46 1 + 2 + 3 + ... + 2000 by Gauss' method
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RESPONSE --> no idea
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19:40:21 ** Pair up the first and last, second and second to last, etc.. You'll thus pair up 1 and 2000, 2 and 1999, 3 and 1998, etc.. Each pair of numbers totals 2001. Since there are 2000 numbers there are 1000 pairs. So the sum is 2001 * 1000 = 2,001,000 **
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RESPONSE --> ok
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19:43:28 Query 1.1.57 142857 * 1, 2, 3, 4, 5, 6. What happens with 7? Give your solution to the problem as stated in the text.
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RESPONSE --> 142857 285714 428571 571428 714285 857142 all contain the same numbers in different orders x7=999999
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19:43:35 ** Multiplying we get 142857*1=142857 142857*2= 285714 142857*3= 428571 142857*4=571428 142857*5= 714285 142857*6=857142. Each of these results contains the same set of digits {1, 2, 4, 5, 7, 8} as the number 1428785. The digits just occur in different order in each product. We might expect that this pattern continues if we multiply by 7, but 142875*7=999999, which breaks the pattern. **
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RESPONSE --> ok
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19:44:00 What does this problem show you about the nature of inductive reasoning?
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RESPONSE --> that it can lead to false assumptions
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19:44:10 ** Inductive reasoning would have led us to expect that the pattern continues for multiplication by 7. Inductive reasoning is often correct it is not reliable. Apparent patterns can be broken. **
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RESPONSE --> ok
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˼ڪ۟N assignment #007 wO}nShwzmнK̹Ѓ Liberal Arts Mathematics I 10-03-2006
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19:45:04 Query 1.2.6 seq 2, 51, 220, 575, 1230, 2317 ... by successive differences
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RESPONSE --> 4392
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19:46:02 ** If the sequence is 2, 57, 220, 575, 1230, 2317, ... then we have: 2, 57, 220, 575, 1230, 2317, # 3992 55, 163, 355, 655, 1087, # 1675 108, 192, 300, 432, # 588 84, 108, 132, # 156 24, 24, The final results, after the # signs, are obtained by adding the number in the row just below, in the following order: Line (4) becomes 132+24=156 Line (3) becomes 432+156=588 Line (2) becomes 1087+588=1675 Line (1) becomes 2317+1675=3992 The next term is 3992. **
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RESPONSE --> ok
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19:46:51 1.2.18 1^2 + 1 = 2^2 - 2; 2^2 + 2 = 3^2 - 3; 3^2 + 3 = etc.
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RESPONSE --> 4^2 +4= 5^2-5 20=20 true
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19:46:55 ** The next equation in the sequence would be 4^2 + 4 = 5^2 - 5 The verification is as follows: 4^2 + 4 = 5^2 - 5 simplifies to give you 16 + 4 = 25 - 5 or 20 = 20 **
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RESPONSE --> ok
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19:47:57 1.2.30 state in words (1 + 2 + ... + n ) ^ 2 = 1^3 + 2^3 + ... + n^3
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RESPONSE --> not sure of the question
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19:48:04 ** the equation says that the square of the sum of the first n counting numbers is equal to the sum of their cubes **
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RESPONSE --> ok
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19:50:25 1.2.36 1 st triangular # div by 3, remainder; then 2d etc. Pattern.
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RESPONSE --> .3repeating 1,2, .30repeating, 5, 7, 9.3repeating two numbers followed by a repeating
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19:51:35 ** The triangular numbers are 1, 3, 6, 10, 15, 21, . . . . We divide these by 3 and get the sequence of remainders. When you divide 1 by 3 you get 0 with remainder 1 (3 goes into 1 zero times with 1 left over). 3 divided by 3 gives you 1 with remainder 0. 6 divded by 3 is 2 with remainder 0. 10 divided by 3 is 3 with remainder 1. Therefore the remainders are 1,0,0,1,0,0. It turns out that the sequence continues as a string of 1,0,0 's. At this point that is an inductive pattern, but remmeber that the sequence of triangular numbers continues by adding successively larger and larger numbers to the members of the sequence. Since the number added always increases by 1, and since every third number added is a multiple of 3, is isn't too difficult to see how the sequence of remainders comes about and to see why it continues as it does. COMMON ERROR: .3333333,1,2,3.3333333,etc. INSTRUCTOR CORRECTION: You need the remainders, not the decimal equivalents. When you divide 1 by 3 you get 0 with remainder 1 (3 goes into 1 zero times with 1 left over). 3 divided by 3 gives you 1 with remainder 0. 6 divded by 3 is 2 with remainder 0. 10 divided by 3 is 3 with remainder 1. Therefore the remainders are 1,0,0,1,0,0 and the sequence continues as a string of 1,0,0 's. COMMON ERROR: 1/3, 1, 2, 3 1/3 CORRECTION: These are the quotients. You need the remainders. If you get 1/3 that means the remainder is 1; same if you get 3 1/3. If you just getting whole number (like 1 or 2 in your calculations) the remainder is 0. In other words, when you divide 1 by 3 you get 0 with remainder 1 (3 goes into 1 zero times with 1 left over). 3 divided by 3 gives you 1 with remainder 0. 6 divded by 3 is 2 with remainder 0. 10 divided by 3 is 3 with remainder 1. The remainders form a sequence 1,0,0,1,0,0 and the sequence continues as a string of 1,0,0 's. **
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RESPONSE --> ok
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19:52:37 1.2.48 use formula to find the 12 th octagonal number. Explain in detail how you used the formula to find this number.
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RESPONSE --> 408 using On=n(6n-4)/2
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19:52:46 ** The pattern to formulas for triangular, square, pentagonal, hexagonal, heptagonal and octagonal numbers is as follows: Triangular numbers: n / 2 * [ n + 1 ] note that this is the same as Gauss' formula Square numbers: n / 2 * [ 2n + 0 ] or just n^2 Pentagonal #'s: n / 2 * [ 3n - 1 ] Hexagonal #'s: n / 2 * [ 4n - 2 ] Heptagonal #'s: n / 2 * [ 5n - 3 ] Octagonal #'s: n / 2 * [ 6n - 4 ] The coefficient of n in the bracketed term starts with 1 and increases by 1 each time, and the +1 in the first bracketed term decreases by 1 each time. You will need to know these formulas for the test. The last formula is for octagonal numbers. To get n = 12 octangonal number use n/2 * [ 6n - 4 ] to get 12 / 2 * [ 6 * 12 - 4 ] = 6 * 68 = 408. **
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RESPONSE --> ok
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}ޜׂD֜} assignment #011 wO}nShwzmнK̹Ѓ Liberal Arts Mathematics I 10-03-2006
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19:54:11 Query 1.4.24 1 to any power is what?
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RESPONSE --> 1
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19:54:15 ** The meaning is 1^2, 1^3, 1^4. We take a power of the base. The result is always 1. **
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RESPONSE --> ok
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19:55:03 Query 1.4.30 sqr of neg gives what?
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RESPONSE --> o,x.x
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19:55:17 ** The square root of a negative will often give you an error (what can you square to get a negative, since any number squared is positive?), but on certain calculators it gives a complex number (actually two complex numbers). These are not real numbers; for the purposes of this course there is no real square root of a negative number. There is no real number that can be squared to give a negative. If you square a negative number you get a negative times a negative, which is positive. If you square a positive number you get a positive number. So a negative number has no real square root. **
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RESPONSE --> ok
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19:55:30 Query 1.4.42 drawer has 18 compartments; how many drawers to hold 204 tapes?
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RESPONSE --> 12
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19:55:35 ** 204 / 18 = 11 with remainder 6. If we had 11 drawers they would hold all but 6 of the tapes. The leftover tapes also have to go into a drawer, so we need a 12th drawer. **
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RESPONSE --> ok
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y~}UЯͤ assignment #008 wO}nShwzmнK̹Ѓ Liberal Arts Mathematics I 10-03-2006
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20:03:08 1.3.6 9 and 11 yr old hosses; sum of ages 122. How many 9- and 11-year-old horses are there?
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RESPONSE --> cannot find question in book but 45 9 year old and 77 11 year old horses
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20:03:19 ** If there was one 11-year-old horse the sum of the remaining ages would have to be 122 - 11 = 111, which isn't divisible by 9. If there were two 11-year-old horses the sum of the remaining ages would have to be 122 - 2 * 11 = 100, which isn't divisible by 9. If there were three 11-year-old horses the sum of the remaining ages would have to be 122 - 3 * 11 = 89, which isn't divisible by 9. If there were four 11-year-old horses the sum of the remaining ages would have to be 122 - 4 * 11 = 78, which isn't divisible by 9. If there were five 11-year-old horses the sum of the remaining ages would have to be 122 - 5 * 11 = 67, which isn't divisible by 9. The pattern is 122 - 11 = 111, not divisible by 9 122 - 2 * 11 = 100, not divisible by 9 122 - 3 * 11 = 89, not divisible by 9 122 - 4 * 11 = 78, not divisible by 9 122 - 5 * 11 = 67, not divisible by 9 122 - 6 * 11 = 56, not divisible by 9 122 - 7 * 11 = 45, which is finally divisible by 9. Since 45 / 9 = 5, we have 5 horses age 9 and 7 horses age 11. **
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RESPONSE --> ok
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20:05:08 Query 1.3.10 divide clock into segments each with same total
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RESPONSE --> segment 1= 11,12,1,2 seg 2=10,9,3,4 seg 3= 8,7,6,5, all = 26
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20:05:14 ** The total of all numbers on the clock is 78. So the numbers in the three sections have to each add up to 1/3 * 78 = 26. This works if we can divide the clock into sections including 11, 12, 1, 2; 3, 4, 9, 10; 5, 6, 7, 8. The numbers in each section add up to 26. To divide the clock into such sections the lines would be horizontal, the first from just beneath 11 to just beneath 2 and the second from just above 5 to just above 8. Horizontal lines are the trick. You might have to draw this to see how it works. **
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RESPONSE --> ok
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20:06:09 Query 1.3.18 M-F 32 acorns each day, half of all acorns eaten, 35 acorns left after Friday
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RESPONSE --> no idea
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20:07:07 ** You have to work this one backwards. If they were left with 35 on Friday they had 70 at the beginning (after bringing in the 32) on Friday, so they had 70 - 32 = 38 at the end on Thursday. So after bringing in the 32 they had 2 * 38 = 76 at the beginning of Thursday, which means they had 76 - 32 = 44 before the 32 were added. So they had 44 Wednesday night ... etc. **
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RESPONSE --> oh ok
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20:07:41 Query 1.3.30 Frog in well, 4 ft jump, 3 ft back.
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RESPONSE --> 20 days 1 foot per day
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20:09:52 ** COMMON ERROR: 20 days CORRECTION: The frog reaches the 20-foot mark before 20 days. On the first day the frog jumps to 4 ft then slides back to 1 ft. On the second day the frog therefore jumps to 5 ft before sliding back to 2 ft. On the third day the frog jumps to 6 ft, on the fourth to 7 ft., etc. Continuing the pattern, on the 17th day jumps to 20 feet and hops away. The maximum height is always 3 feet more than the number of the day, and when max height is the top of the well the frog will go on its way. **
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RESPONSE --> haha ok
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20:12:18 ** Divide the coins into two piles of 4. One pile will tip the balance. Divide that pile into piles of 2. One pile will tip the balance. Weigh the 2 remaining coins. You'll be able to see which coin is heavier. **
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RESPONSE --> 6 i dime 5 pen 15pen 3 nickles 1 dime 1 nickle 10pen 1 nickle 2 nickle 5 pen.
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}gݍd} assignment #012 wO}nShwzmнK̹Ѓ Liberal Arts Mathematics I 10-03-2006
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20:15:05 Query 3.1.10 Mary is top grossing film. Is this a statement?
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RESPONSE --> yes
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20:17:17 ** A sentence is a statement if it is true or false. Otherwise it is not a statement. 'There goes a big one' is not a statement because the word 'big' is open to interpretation so is not a statement. 'There are 3.87 * 10^89 particles in the universe at this instant' is a statement: it is either true or it isn't, though we don't know enough to tell which. The gross receipts for a film can be regarded as hard facts--unlike opinions on whether a film is good, or artistic. If 'a top grossing film' is defined as, say, a top-10 film in gross receipts, then we could ascertain whether it is true or false and we would have a the statement. However, 'top grossing' isn't defined here--does it mean one of the top three for the week, top 10 for the year, or what?--and for that reason we can't decide for sure whether it is true or false. So this sentence couldn't be regarded as a statement. **
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RESPONSE --> ok
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20:17:38 Query 3.1.12 sit up and behave Is this a statement?
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RESPONSE --> no, command
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20:17:43 ** This is not a statement. It is a command. You could evaluate the truth of the statement 'you sat up', but not the truth of the command to sit up. **
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RESPONSE --> ok
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20:19:00 Query 3.1.18 Is this a compound statement or not: 'calif or bust'.
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RESPONSE --> yes
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20:19:11 ** The meaning of the phrase is actually 'we're gonna get to California or we're gonna break down trying', in which context it is a compound statement involving 'or'. **
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RESPONSE --> ok
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20:19:28 Query 3.1.30 negate 'some people have all the luck
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RESPONSE --> some people do not have luck
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20:20:19 ** The negation would be 'all people don't have all the luck', which means 'nobody has all the luck'. The negation of 'some do' is indeed 'all do not', which is the same as 'none do'. The negation of 'all do' is 'some do not'. The negation of 'none do' is 'some do'. COMMON ERROR: Not everyone has all the luck, or equivalently some people do not have all the luck. This is not incompatible with the original statement, and the negation must be incompatible. Both would be true if some do have all the luck and some don't. **
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RESPONSE --> ok
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20:20:57 Query 3.1.42 p: she has green eyes q: he is 48. What is the statement (p disjunction q)?
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RESPONSE --> she has green eyes or he is 48 years old.
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20:21:02 ** The statement is 'She has green eyes or he is 48 yrs. old' **
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RESPONSE --> ok
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20:21:54 Query 3.1.48 What is the statement -(p disjunction q)
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RESPONSE --> It is not the case that she has green eyes or he is 48 years old.
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20:22:00 ** The correct translation is 'It is not the case that she has green eyes or he is 48 yrs. old'. An equivalent statement, using deMorgan's Laws, would be 'she doesn't have green eyes and he is 48 years old' COMMON ERROR: She doesn't have green eyes or he is not 48 years old. This statement negates p V q as ~p V ~q, which is not correct. The negation of p V q is ~p ^ ~q. **
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RESPONSE --> ok
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20:24:26 Query 3.1.54 Jack plays tuba or Chris collects videos, and it is not the case that both are so
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RESPONSE --> p(down arrow) q+~(p^q)
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20:24:43 ** The statement 'jack plays or Chris collects' is symbolized by (p U q). The statement that it is not the case that both are so is symbolized ~(p ^ q). The entire statement is therefore (p U q) ^ ~(p ^ q).**
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RESPONSE --> ok
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20:25:30 3.1.60 true or false: there exists an integer that is not a rational number.
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RESPONSE --> true
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20:25:40 ** A rational number is a number that can be written as p / q, with p and q both integers. Examples are 2/3, -5489/732, 6/2, etc.. Other examples could be 5/1, 12/1, -26/1; these of course reduce to just 5, 12, and -26. The point is that any integer can be written in this form, with 1 in the denominator, so any integer is in fact also a rational number. Thus there is no integer that is not a rational number, and the statement is false. **
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RESPONSE --> ok
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20:26:15 Query 3.1.66 true or false: each rat number is a positive number
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RESPONSE --> false
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20:26:19 ** This is false, and to prove it you need only give an example of a rational number that is negative. For example, -39/12 is a rational number (integer / integer) and is negative. **
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RESPONSE --> ok
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z^ͱҾDݗHڿ assignment #012 wO}nShwzmнK̹Ѓ Liberal Arts Mathematics I 10-03-2006"