Prove the sum of the even Fibonacci numbers

Question

Let $f_n$ denote the $nth$ Fibonacci number. Prove that

$f_2\:+\:f_4\:+...+f_{2n}=f_{2n+1}-1$

I am having trouble proving this. I thought to use induction as well as Binet's formula where,

$f_n=\frac{\tau^2-\sigma^2}{\sqrt5}$ where $\tau=\frac{1+\sqrt5}{2}$ and $\sigma=\frac{-1}{\tau}$.

Can someone give me a hand?

Thanks!

Perhaps it's easier in the form $$f_{2n+1} = f_{2n}+f_{2n-2}+\cdots+f_6+f_4+f_2+f_1$$ Just keep expanding the recurrence relation for the odd-indexed numbers! (Of course, this would formally map to an induction proof). — hmakholm left over Monica
– hmakholm left over Monica, Commented Feb 22, 2015 at 1:57

kobe · Accepted Answer · 2015-02-22 01:59:44Z

Using the recurrence relation for $f_n$ we find

\begin{align}f_2 + f_4 + \cdots + f_{2n} = (f_3 - f_1) + (f_5 - f_3) + \cdots + (f_{2n+1} - f_{2n-1}), \end{align}

which telescopes to $f_{2n+1} - f_1 = f_{2n+1} - 1$.

NoName · Accepted Answer · 2015-02-22 02:03:16Z

This can be proven with induction. If we're saying the sequence starts $1,1,2,\ldots$ then clearly $f_2 = f_3 - 1$.

Now assume $f_2\:+\:f_4\:+\ldots+f_{2n}=f_{2n+1}-1$. Add $f_{2n+2}$ to both sides. Then

$$f_2\:+\:f_4\:+...+f_{2n} + f_{2n+2}= f_{2n+2}+f_{2n+1}-1$$

And since $f_{2n+2}+f_{2n+1} = f_{2n+3}$, the result follows.

2 Answers 2