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Help. I need help with a practice problem I am doing for while studying for my exam. I started working on the problem. I used mathematical induction. We are given: F_(n+1)=F_n +F_(n+1) and F_1=F_2=1 Step 1: I prove the base case: n=1

f_1+1 f_1+1 = (f_1)^2 + (-1)^1 1(0)=1^2 + (-1)^1 0=0 Check, the base case.

Assume it is true for all k=n. That is (F_k+1)(F_k-1)=(F_k)^2 + (-1)^k.

Now, for k=n+1, we have

F_(n+1)+1 F_(n+1)-1 = (f_n+1)^2 + (-1)^n+1 =>

F_n+1F_n = (F_n+1)^2 + (-1)^(n+1)

LHS: F_(n+2)F_n = (F_(n+1) + F_n)Fn

= F_n+1 (F_n) + F_n(F_n)

Idk where to go know or if my work is incorrect.

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Courtney J
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1 Answers1

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Using what you did so far -- which seems correct -- you wish to show that

$F_{n+1}^2+(-1)^{n+1}=F_{n+1}F_n+F_nF_n$.

Since $-(-1)^{n+1}=(-1)^n$, that is $F_{n+1}(F_{n+1}-F_n)=F_n^2+(-1)^n,$

which holds because of the inductive hypothesis $F_{n+1}F_{n-1}=F_n^2+(-1)^n$

and $F_{n+1}-F_n=F_{n-1}$.

J. W. Tanner
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