Fibonacci Numbers

[JellyFish]

I am looking for a bit of help completing 2 induction proofs of some Fibonacci identities.

1. (F[sub:8ov6t2og]0[/sub:8ov6t2og] )^2 + (F[sub:8ov6t2og]1[/sub:8ov6t2og])^2 + (F[sub:8ov6t2og]2[/sub:8ov6t2og] )^2 +. . . + (F[sub:8ov6t2og]n[/sub:8ov6t2og] )^2 = F[sub:8ov6t2og]n[/sub:8ov6t2og]*F[sub:8ov6t2og]n+1[/sub:8ov6t2og] . For n?0.
Thus far,
Basis is n=0, F[sub:8ov6t2og]0[/sub:8ov6t2og]^2 = 0 F[sub:8ov6t2og]0[/sub:8ov6t2og]*F[sub:8ov6t2og]1[/sub:8ov6t2og] = 0.

Induction hypothesis is n=k, (F[sub:8ov6t2og]0[/sub:8ov6t2og] )^2 + (F[sub:8ov6t2og]1[/sub:8ov6t2og])^2 + (F[sub:8ov6t2og]2[/sub:8ov6t2og] )^2 +. . . + (F[sub:8ov6t2og]k[/sub:8ov6t2og] )^2 = F[sub:8ov6t2og]k[/sub:8ov6t2og]*F[sub:8ov6t2og]k+1[/sub:8ov6t2og]

Induciton step is n=k+1, (F[sub:8ov6t2og]0[/sub:8ov6t2og] )^2 + (F[sub:8ov6t2og]1[/sub:8ov6t2og])^2 + (F[sub:8ov6t2og]2[/sub:8ov6t2og] )^2 +. . . + (F[sub:8ov6t2og]k[/sub:8ov6t2og] )^2 + F[sub:8ov6t2og]k+1[/sub:8ov6t2og] )^2
= F[sub:8ov6t2og]k+1[/sub:8ov6t2og]*F[sub:8ov6t2og]n(K+1)+1[/sub:8ov6t2og]

Then Substituting what we know from n=k, F[sub:8ov6t2og]k[/sub:8ov6t2og]*F[sub:8ov6t2og]k+1[/sub:8ov6t2og] + F[sub:8ov6t2og]k+1[/sub:8ov6t2og] )^2 = F[sub:8ov6t2og]k+1[/sub:8ov6t2og]*F[sub:8ov6t2og]n(K+1)+1[/sub:8ov6t2og]

And this is where I am not quit sure what to do?

Then 2.
F[sub:8ov6t2og]1[/sub:8ov6t2og] + F[sub:8ov6t2og]3[/sub:8ov6t2og] + F[sub:8ov6t2og]5[/sub:8ov6t2og] +. . . + F[sub:8ov6t2og]2n+1[/sub:8ov6t2og] = F[sub:8ov6t2og]2n+2[/sub:8ov6t2og] for n ?0.
So far,

Basis: n=0, F[sub:8ov6t2og]2*0+1[/sub:8ov6t2og] = 1 F[sub:8ov6t2og]2*0+2[/sub:8ov6t2og] = 1

Induction Hypothesis is n =k, F[sub:8ov6t2og]1[/sub:8ov6t2og] + F[sub:8ov6t2og]3[/sub:8ov6t2og] + F[sub:8ov6t2og]5[/sub:8ov6t2og] +. . . + F[sub:8ov6t2og]2k+1[/sub:8ov6t2og] = F[sub:8ov6t2og]2k+2[/sub:8ov6t2og]

Induction Step is n=k+1, F[sub:8ov6t2og]1[/sub:8ov6t2og] + F[sub:8ov6t2og]3[/sub:8ov6t2og] + F[sub:8ov6t2og]5[/sub:8ov6t2og] +. . . + F[sub:8ov6t2og]2(k+1)+1[/sub:8ov6t2og] = F[sub:8ov6t2og]2(k+1)+2[/sub:8ov6t2og]
The sub what we know from n=k,
F[sub:8ov6t2og]2k+2[/sub:8ov6t2og] + F[sub:8ov6t2og]2(k+1)+1[/sub:8ov6t2og] = F[sub:8ov6t2og]2(k+1)+2[/sub:8ov6t2og]

Again, Im just not sure where to go from.

Thank you

 

[daon]

First one, base case: \(\displaystyle F_0^2 = 0^2 = 0 = (0)(1) = F_0F_1\)

Your inductive step is also carelessly written. You have parentheses randomly placed, things aren't squared where they should be, etc.

You ASSUME: \(\displaystyle \sum_{k=0}^{n}F_k^2 = F_nF_{n+1}\)

You need to PROVE: \(\displaystyle \sum_{k=0}^{n+1}F_k^2 = F_{n+1}F_{n+2}\)

Start with what you ASSUME, add \(\displaystyle F_{n+1}^2\) to both sides, you get:

\(\displaystyle \sum_{k=0}^{n}F_k^2 + F_{n+1}^2 = F_nF_{n+1} + F_{n+1}^2\)

i.e.

\(\displaystyle \sum_{k=0}^{n+1}F_k^2 = F_nF_{n+1} + F_{n+1}^2\)

Try to finish from there... and try the second question in a similar way.