course

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

.

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

006. Sequences and Patterns

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Question: `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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Your solution: 16 because each element increases by 1

Confidence rating: 3

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Given Solution:

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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Self-critique (if necessary):

Self-critique Rating:

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Question: `q002. Find the likely next two elements of the sequence 1, 2, 4, 8, 15, 26, ... .

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 42, 64 it’s a little hard to explain my reasoning without pencil and paper but from one to two there is a change of 1 and from 2 to 4 there is a change of 2 and from 1 to 2 there is a change of one and so fourth.

Confidence rating: 3

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Given Solution:

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 orig69inal 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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Self-critique (if necessary): I have gone through the work several times and I am still coming up with 64 instead of 68

you are correct

Self-critique Rating:

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Question: `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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Your solution: the next number would be 16 because each element is double the previous element

Confidence rating: 3

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Given Solution:

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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Self-critique (if necessary):

Self-critique Rating:

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Question: `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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Your solution: The ratio is 1.5 1.5 1.5 so the next number in the set is 162

Confidence rating:

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Given Solution:

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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Self-critique (if necessary):

Self-critique Rating:

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Question: `q005. Find the sequence of ratios for the sequence 1, 2, 3, 5, 8, 13, 21... , and estimate the next element of the sequence.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Confidence rating:

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Given Solution:

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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Self-critique (if necessary):I assumed that there was no pattern and therefore we could not decide what the next number would be I don’t understand how we decided that 34 would be the next number in the sequence or what ratio to use

The pattern of the ratios indicates that the ratio will be between 1.615 and 1.625, and also that it will be closer to 1.615. So 1.618 is a good estimate, based on the pattern that has been established.

However either of the ratios 1.615 and 1.625 will still give you a number that rounds to 34.

Self-critique Rating:

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Question: `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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Your solution: yes each element is the sum of the two previous ones

8+13=21

13+21=34

Confidence rating:

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Given Solution:

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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&#This looks good. See my notes. Let me know if you have any questions. &#