Arnold F. answered • 04/12/16

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Freya,

Let P(n) be Σ

^{n}_{i=1}^{ }F_{i}^{2}= F_{n}x F_{n+1}In a proof by induction the first (Basis) step is to show P(1) is true so:

P(1): F

_{1}^{2}= F_{1}x F_{2} 1

^{2}= 1 x 1 is trueNow we assume P(k) is true. (Inductive Hypothesis)

The inductive proof step:

Using the Inductive Hypothesis you want to now show P(k+1) is true:

(1) F

_{1}^{2}+ F_{2}^{2}+ F_{3}^{2}+ ... F_{k}^{2}+ F_{k+1}^{2 }= F_{k}x F_{k+1}+ F_{k+1}^{2}Can you do the rest of the proof from here?