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Using the Fibonacci sequence and induction prove that

$$F_{n-1}F_{n+1}-F_{n}^2 = (-1)^n, \space \space n=1,2,3...$$

My efforts so far:

The basis holds for $n=1$

Induction step: $$F_{n-1}F_{n+1}-F_{n}^2 + F_{n}F_{n+2}-F_{n+1}^2 = (-1)^{n+1}$$ LHS: $$(-1)^n + F_{n}F_{n+2}-F_{n+1}^2 = $$ $$(-1)^n + F_n(F_{n+1}+F_{n})- F_{n+1}(F_n+F_{n-1}) = $$ $$(-1)^n+F_n^2 - F_{n+1}F_{n-1} = $$ $$(-1)^n-(-1)^n=0$$

What am I doing wrong?

DoubleOseven
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1 Answers1

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I don't see why you compute $\;\color{red}{F_{n-1}F_{n+1}-F_{n}^2} + F_{n}F_{n+2}-F_{n+1}^2$. Computing $\; F_{n}F_{n+2}-F_{n+1}^2$ is all you need, and in this case what you did works fine.

Bernard
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  • I thought that is the way induction works. That you have to use the assumption of $n$ to get to $n+1$ – DoubleOseven Feb 19 '16 at 10:49
  • Using the inductive hypothesis does not mean adding the expression for case $n$ to the expression for case $n+1$. – Bernard Feb 19 '16 at 10:52