Open Query 11

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course Mth 151

2/10 3

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

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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 7 questions in this assignment.

Find the likely next element of the sequence 1, 2, 4, 7, 11, ... .

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Your solution:

16

We can assume that the numbers are being added to a sequence of 1,2,3,4…

1+1=2, 2+2=4, 4+3=7, 7+4=11, 11+16

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):ok

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Self-critique Rating:

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

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Your solution:

OK I could not get this one, but now that I know the method I can work out the problem.

We can see that the differences of each element from the next makes a sequence of 1,2,4,7,11… Which we know from the previous question to be obtained by adding each by the sequence 1,2,3,4…. So this equation has two levels. We start at the first level by continuing the sequence 1,2,3,4, and adding 5,6 to the mix. Then the second sequence is 1,2,4,7,11,16,22…. We can then take that sequence and add it to the original in that pattern, so the 26 will be added by 16 to make the next number in the sequence 42. So the sequence becomes 1,2,4,8,15,26,42,68

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 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 + 22 = 64 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):ok

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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:

We can assume this one is by doubling the previous element. So to find the next we simply double 8 to get 16, making the sequence

1, 2, 4, 8, 16

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):ok

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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:

Each element is multiplied by 1.5 in order to get the next. That makes the next number in the sequence equal 162. The sequence is 32, 48, 72, 108, 162 and the ratio is

48/32=1.5, 72/48=1.5, 108/72=1.5, 162/108=1.5

confidence rating #$&*:3

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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):ok

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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.

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Your solution:

The next in the sequence is 34, because each element is the sum of the two before it. The ratio would then be

2/1=2, 3/2= 1.5, 5/3=1.66, 8/5=1.6, 13/8=1.625, 21/13= 1.615, 34/21=1.619

confidence rating #$&*:

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Given Solution:ok

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):ok

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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:

Same addition as before,

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

confidence rating #$&*:3

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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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Question: `q007. What is the pattern of the sequence

... 4 7 11 18 ...

and what are the next three numbers in this sequence?

What positive numbers could precede the 4, how did you figure them out and how do you know you got them all?

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Your solution:

1, 3, 4, 7, 11, 18, 29, 47, 76…

To find the 3 at the end you simply do what has been done before and add the two before the next number in order to find the element that comes next. To find those preceding, you work backwards and subtract the second number by the first in order to get what came before either of them

7-4= 3, 4-3=1

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