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$$\sum_{i = 1}^n F_{i}^2 = F_n F_{n+1}$$

-I solved a similar Fibonacci sequence that was the following: $$\sum_{i = 1}^n F_i = F_{n + 2} - 1$$ But, I am having trouble with this one, any help is appreciated.

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  • $\begingroup$ Have you tried to prove this by induction? $\endgroup$ Jan 31 '16 at 22:50
  • $\begingroup$ This section is on induction proofs, I did the base case but am stuck on the inductive step. $\endgroup$ Jan 31 '16 at 22:51
  • $\begingroup$ Could you edit your question to show us what you have done? $\endgroup$ Jan 31 '16 at 22:52
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By induction: $$F_1^2 + \dots + F_{n+1}^2 = F_n F_{n+1} + F_{n+1}^2 = F_{n+1} (F_n + F_{n+1}) = F_{n+1} F_{n+2}. $$

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Hint: $~F_nF_{n+1}~=~F_n(F_n+F_{n-1})~=~F_n^2+F_nF_{n-1}~=~F_n^2+F_{n-1}(F_{n-1}+F_{n-2})~=$

$=~F_n^2+F_{n-1}^2+F_{n-1}F_{n-2}~=~\cdots$

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