1. ## Prove Conjecture

Prove conjecture for the sum of the squares of the first n Fibonacci numbers

(F1) ^2 + (F2) ^ 2 .... + (Fn) ^2

Thanks for the help!!

2. What is the conjecture?

3. well I'm really trying to prove

f1^2 + F2^2 +.... Fn^2 = (Fn) (Fn+1) by assuming n = K + 1

Thanks for Speedy response

4. This is certainly an inductive proof. But it is not difficult.

We have:

$1^2 = (1)(1)$

$1^2 + 1^2 = (1)(2)$

$1^2 + 1^2 + 2^2 = (2)(3)$

So it looks like the first few cases are true.

Suppose $\sum _{k=1} ^n (F_k)^2 = F_nF_{n+1}$ is true.

Then $\sum _{k=1} ^{n+1} (F_k)^2 = F_nF_{n+1} + (F_{n+1})^2 = (F_n + F_{n+1})(F_{n+1}) = F_{n+2}F_{n+1} = F_{n+1}F_{n+2}$.

And the induction is complete.

5. ahh thanks for the help