course Mth 151 Attached is my query for assignments 6 and 7.Thank you,
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13:21:05 `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 --> The likely next element in the sequence of 1, 2, 4, 11 is 16. To get this we 1+1 to get 2, then 2+2 to get 4, then 4+3 to get 7, then 7+4 to get 11, then 11+5 to get 16. confidence assessment: 3
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13:21:28 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 --> Correct answer. self critique assessment: 3
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13:28:56 `q002. Find the likely next two elements of the sequence 1, 2, 4, 8, 15, 26, ... .
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RESPONSE --> The likely next two elements in the sequence would be 42, 64. We get this by looking at the previous sequence 1,2,4,7,11,16 knowing that each number was added by 1,2,3,4,5, carrying this on then 11+5 would give us 16 and adding 16 to 26 gives us the next element in this sequence of 42 and then for the second number back to the previous sequence 16+6=22 and 22 added to 42 will give us 64. confidence assessment: 2
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13:34:50 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 that there is an error in typing in the solution to the problem in the last paragraph where the number 22 has been changed to 26 in the same sentence. The answers I got were 42 and 64. self critique assessment: 3
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13:37:14 `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 --> The next likely element for this sequence 1,2,4,8 could be 16 by using multiples of 2 and multipling 8 by 2= 16. confidence assessment: 2
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13:37:30 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 --> Correct answer. self critique assessment: 3
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13:49:44 `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 --> The next element for the sequence 32,48,72,108 is 162 and the sequence of ratios for the sequence is 2,2,2,2 since 32/16=2 ; 48/24=2; 72/36=2; 108/54=2 and 162/81=2. confidence assessment: 2
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13:52:31 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 --> I got the correct answer of 162 but did not use the correct way to get the sequence of ratios. I divided the wrong numbers into each other. self critique assessment: 2
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14:01:40 `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 --> The next element in the sequence is going to be 34. I got this answer by 1+1=2; 2+1=3; 2+3=5; 3+5=8; 5+8=13; 8+13=21; 21+13=34. When I did the ratio as with the previous problem I got all different numbers; 2,1.5, 1.666, 1.6, 1.625, 1.615. confidence assessment: 1
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14:06:14 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 --> I had the correct answer and after reading your explanation I have a better understand of the ratio. I thought that by getting all these different numbers for the ratios that I was incorrect. self critique assessment: 2
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14:16:54 `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 --> The pattern for this sequence is 1+0=1; 1+1=2; 2+1=3; 3+2=5; 5+3=8; 8+5=13; 13+8=21; 21+13=34; 34+21=55; 55+34=89; 89+55=144. So the next three elements would be 55,89,144. confidence assessment: 3
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14:17:16 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 --> Correct answer. self critique assessment: 3
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Eϭx̻őCywפ assignment #007 007. Triangular, Square, Pentagonal Numbers Liberal Arts Mathematics I 09-09-2008
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20:22:39 `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 --> The original triangle had 3 points and the new triangle has 6 points marked. confidence assessment: 3
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20:23:36 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 --> Correct answer. self critique assessment: 3
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20:30:08 `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 --> There are 10 total end points marked on the whole figure and 8 on the new figure. confidence assessment: 2
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20:30:45 You added the two new endpoints when you extended the sides. You then should have marked two new points on the new third side, so that each side contains 4 points including its endpoints. Your figure will now contain 10 marked points.
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RESPONSE --> Correct on the total endpoints for new figure. self critique assessment: 2
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20:32:40 `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 --> There are 15 points on the new triangle. confidence assessment: 3
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20:33:13 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 --> Correct. self critique assessment: 3
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20:35:25 `q004. Continue the process for one more step. How many points do you have in the new triangle?
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RESPONSE --> There are 21 points in the new triangle. confidence assessment: 3
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20:36:12 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 --> Correct answer with what I have. self critique assessment: 3
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20:39:41 `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 --> The probable next element in this sequence will be 28 and we get this by adding 21+7. The pattern for this sequence is 3+3=6, 6+4=10, 10+5=15, 15+6=21, so 21+7=28. confidence assessment: 3
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20:40:01 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 --> Correct. self critique assessment: 3
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20:44:35 `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 --> By extending the line segments of the triangle and marking the endpoints to show that there are 28 points total in the new triangle that are marked. confidence assessment: 2
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20:44:51 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 --> self critique assessment: 3
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20:54:23 `q007. How do you know this sequence will continue in this manner?
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RESPONSE --> For each time you extend the length of the side you add an extra endpoint on that sides which in turn will give you the next number in the previous sequence. confidence assessment: 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 --> Correct. self critique assessment: 3
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