Proof by Induction: Alternating Sum of Fibonacci Numbers [duplicate]
Solution 1:
Your reasoning is sound, but your induction hypothesis is a bit wrong. It should be something like this:
Assume that it holds for $n = k$. We then have $$ f_0 -f_1 + \cdots -f_{2k-1} +f_{2k} = f_{2k-1} -1 $$ I will now show that $$ f_0 -f_1 + \cdots -f_{2k-1} +f_{2k} -f_{2k + 1} + f_{2k + 2} = f_{2k+1} -1 $$
If you solve that, then it will be proven.
Solution 2:
You get the inductive step by replacing $n$ everywhere with $n+1$. So the alternating sum should go up to $2(n+1)$, not $2n+1$.