Prove conjecture for the sum of the squares of the first n Fibonacci numbers
(F1) ^2 + (F2) ^ 2 .... + (Fn) ^2
Thanks for the help!!
This is certainly an inductive proof. But it is not difficult.
We have:
$\displaystyle 1^2 = (1)(1)$
$\displaystyle 1^2 + 1^2 = (1)(2)$
$\displaystyle 1^2 + 1^2 + 2^2 = (2)(3)$
So it looks like the first few cases are true.
Suppose $\displaystyle \sum _{k=1} ^n (F_k)^2 = F_nF_{n+1}$ is true.
Then $\displaystyle \sum _{k=1} ^{n+1} (F_k)^2 = F_nF_{n+1} + (F_{n+1})^2 = (F_n + F_{n+1})(F_{n+1}) = F_{n+2}F_{n+1} = F_{n+1}F_{n+2}$.
And the induction is complete.