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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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