Query 26

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course Mth 151

4/5 10:40

If your solution to stated problem does not match the given solution, you should self-critique per instructions athttp://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.

027. `query 27

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Question: `q5.4.6 Give the approximate value of Golden Ratio to thousandth and show how you obtained your result.

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Your solution: 1.618

I looked in the book.

confidence rating #$&*: 3

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Given Solution:

`a** The Golden Ratio is [ 1+`sqrt(5) ] /2

[ 1+`sqrt(5) ] /2=[ 1+2.2361 ] /2=3.2361/2=1.618 **

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Self-critique (if necessary):

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Self-critique Rating: 3

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Question: `q5.4.12 2^3 + 1^3 - 1^3 = 8; 3^3 + 2^3 - 1^3 = 34; 5^3 + 3^3 - 2^3 = 144; 8^3 + 5^3 - 3^3 = 610.

What are the next two equations in this sequence?

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Your solution: 13^3+8^3-5^3=2584

21^3+13^3 - 8^3=10946

confidence rating #$&*: 3

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Given Solution:

`a** The numbers 1, 1, 2, 3, 5, 8, ... are the Fibonacci numbers f(1), f(2), ... The left-hand sides are

f(3)^3 + f(2)^3 - f(1)^3,

f(4)^3 + f(3)^3 - f(2)^3,

f(5)^3 + f(4)^3 - f(3)^3,

f(6)^3 + f(5)^3 - f(4)^3 etc..

The right-hand sides are f(5) = 8, f(8) = 34, f(11) = 144, f(14) = 610. So the equations are

f(3)^3 + f(2)^3 - f(1)^3 = f(5)

f(4)^3 + f(3)^3 - f(2)^3 = f(8)

f(5)^3 + f(4)^3 - f(3)^3 = f(11)

f(6)^3 + f(5)^3 - f(4)^3 = f(14)

etc..

The next equation would be

f(7)^3 + f(6)^3 - f(5)^3 = f(17).

Substituting f(7) = 13, f(6) = 8 and f(5) = 5 we get

13^3 + 8^3 - 5^3 = f(17). The left-hand side gives us result 2584, which is indeed f(17), so the pattern is verified in this instance. **

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Self-critique (if necessary):

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Self-critique Rating: 3

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Question: `q5.4.18 show whether F(p+1) or F(p-1) is divisible by p.

Give your solution to this problem.

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Your solution: I do not understand how to find the answer. I will have to look at the given solution.

confidence rating #$&*: 0

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Given Solution:

`a** For p=3 we get f(p-1) = f(2) = 1 and f(p+1) = f(4)= 3; f(p+1) = f(4) = 3 is divisible by p, which is 3 So the statement is true for p = 3.

For p=7 we get f(p-1) = f(6) = 8 and f(p+1) = f(8) = 21; f(p+1) = 21 is divisible by p = 7. So the statement is true for p = 7.

For p = 11 we get f(p-1) = f(10)= 55 and f(p+1) = f(12) = 144. f(p-1) = 55 is divisible by p = 11. So the statement is true for p = 11.

So the conjecture is true for p=3, p=7 and p=11.**

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Self-critique (if necessary):I still don’t get it. ???What value does f have???

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Self-critique Rating:0

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F is the name of a function.

F(1) is the first Fibonacci number

F(2) is the second.

F(3) is the third.

F(4) is the fourth.

etc..

So

F(1) = 1

F(2) = 1

F(3) = 2

F(4) = 3

F(5) = 5

F(6) = 8

F(7) = 13

Now, for example, you know that the eigth Fibonacci number is the sum of the sixth and the seventh. so the eighth Fibonacci number is 8 + 13 = 21. So F(8) = 21.

In terms of the F notation we could write

F(8) = F(6) + F(7).

Similarly we know that the ninth number is the series is the sum of the seventh and the eighth, so

F(9) = F(7) + F(8).

We know F(7) = 13, and we just figured out that F(8) = 21.

So

F(9) = 13 + 21 = 34.

That is, the ninth number in the Fibonacci series is 34.

Now, if p = 7 then

F(p) = F(7) = 13.

We don't really care about F(p); the question asks about F(p - 1) and F(p + 1). Since p = 7, we know that p - 1 = 6 and p + 1 = 8, so

F(p - 1) = F(6) = 8

and

F(p + 1) = F(8) = 21.

The question asks whether F(p + 1) or F(p - 1) is divisible by p. Remembering that p = 3 we can write

F(p - 1) / p = 8 / 3

and

F(p + 1) / p = 21 / 3.

3 doesn't evenly divide 8, so the first statement isn't true.

3 does divide 21 so the second statement is true.

For p = 3, we conclude that F(p + 1) is divisible by p.

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You could apply similar reason to see how this works out for p = 7 or p = 11.

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Question: `q5.4.24 Lucas sequence: L2 + L4; L2 + L4 + L6; etc.. Give your solution to this problem as stated in your text.

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Your solution: I’m sorry. What is the Lucas sequence? I’m probably looking over something.

confidence rating #$&*: 0

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Given Solution:

`a** The Lucas sequence is 1 3 4 7 11 18 29 47 76 123 199 etc.

So

L2 + L4 = 3 + 7 = 10;

L2 + L4 + L6 = 3 + 7 + 18 = 28;

L2 + L4 + L6 + L8 = 3 + 7 + 18 + 47 = 75, and

L2 + L4 + L6 + L8 + L10 = 3 + 7 + 18 + 47 + 123 = 198.

Note that 10 is 1 less than 11, which is L5; 27 is 1 less than 28, which is L7; and 198 is 1 less than 199, which is L9.

So L2 + L4 = L5 - 1, L2 + L4 + L6 = L7 - 1, etc..

So we can conjecture that the sum of a series of all evenly indexed members of the Lucas sequence, starting with index 2 and ending with index 2n, is 1 less than member 2n+1 of the sequence. **

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Self-critique (if necessary): Oh now I feel stupid. I just located the Lucas sequence in my book. I was looking at the wrong page. I understand how this works.

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Self-critique Rating:0

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Far from stupid. You're doing quite well.

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Do remember what the Lucas sequence is, especially when you take the test.

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Self-critique (if necessary):

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Self-critique (if necessary):

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Self-critique rating:

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