Proving the Fibonacci Numbers Using Induction

[rooski]

Homework Statement

use induction to prove

(my formatting is off sorry)
$$\overline{n} \sum \underline{k=1} f _{2k-1} = f_{2n}$$

The Attempt at a Solution

To start we need to show that f3 is valid. So we show that f2 + f1 = f3, which is the case.

The next part is the confusing part for me. Do i have to show a sum ending at n+1 that computes f2k-1 = f2n+1?

 

[hgfalling]

It seems that what you want is:

$$\sum^n_{k=1} f_{2k-1} = f_{2n}$$

So you showed this for n = 3.

Now you need to show that this statement being true for n=j IMPLIES that it is true for n=j+1. So assume that it's true for j, and show that then it is true for j+1.

 

[HallsofIvy]

In other words, assume that
$$\sum_{k=1}^j f_{2k-1}= f_{2j}$$
and use that to prove that
$$\sum_{k= 1}^{j+1} f_{2k-1}= f_{2j+2}$$