assignments 6 7

course Mth 151

?Y???_??????assignment #006

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006. Sequences and Patterns

Liberal Arts Mathematics I

10-02-2007

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00:54:28

`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 next element is 16

1+1=2

2+2 =4

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7+4=11

11+5=16

confidence assessment: 3

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00:57:56

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 -->

?

self critique assessment: 0

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01:03:25

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 did not under stand this problem

self critique assessment: 0

&#

Your response did not agree with the given solution in all details, and you should therefore have addressed the discrepancy with a full self-critique, detailing the discrepancy and demonstrating exactly what you do and do not understand about the given solution, and if necessary asking specific questions (to which I will respond).

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01:04:32

`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 -->

32

confidence assessment: 1

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01:05:51

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 -->

162

self critique assessment: 1

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01:14:01

`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 -->

33

confidence assessment: 3

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01:15:51

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 -->

33+17=50

self critique assessment: 3

You didn't find the sequence of ratios and didn't use that result to estimate the next element. Neither your solution nor you self-critique demonstrates that you understand this idea.

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01:22:03

`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 -->

55,88,138

confidence assessment: 3???????ve???assignment #007

007. Triangular, Square, Pentagonal Numbers

Liberal Arts Mathematics I

10-25-2007

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23:29:19

`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 triangle has 3 points A, B, and C.

The new triangle has 6 points.

confidence assessment: 3

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23:30:08

`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 -->

The new triangle is 4 points

as a whole there is 10 points.

confidence assessment: 3

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23:30:52

`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 -->

The new triangle has 13 points , and as a whole there is 15 points.

confidence assessment: 3

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23:32:16

`q004. Continue the process for one more step. How many points do you have in the new triangle?

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RESPONSE -->

The new triangles has 15 points and as a whole there is 21 points

confidence assessment: 3

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23:35:06

`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 diff. is from 3 to6 is goes up 3. from 6 to 10 it goes up 4. from 10 to 15 it goes up 5. from 15 to 21 it goes up 6.

If you go up the 7

then the next number in the seq is 28

confidence assessment: 3

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23:36:15

`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 -->

you will add 2 points to the sides and the total # of the triangle will be 7 new points.

confidence assessment: 2

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23:37:15

`q007. How do you know this sequence will continue in this manner?

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RESPONSE -->

when you increase the sides by 1, Then the whole triangle increases.

confidence assessment: 3

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Good answers on the last few questions.

However on many questions you didn't answer with sufficient detail and didn't include detailed self-critiques.