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Prove the following statements with either induction, strong induction or proof by smallest counterexample. Concerning the Fibonacci sequence, prove that k=1nFk2=FnFn+1\sum_{k=1}^{n} F_{k}^{2}=F_{n} F_{n+1}.


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Remember:\text{\underline{Remember:}} The Fibonacci numbers\text{\textcolor{#c34632}{Fibonacci numbers}} are defined to be F1=1,F2=1F_1 = 1 , F_2 = 1 and Fn=Fn1+Fn2F_n = F_{n-1} + F_{n-2} for n>2n > 2.

Proof by Mathematical induction\textbf{Proof by Mathematical induction}

For n=1n=1: L.H.S =k=11Fk2=F12=1=\sum_{k=1}^{1}{F_k}^2={F_1}^2=1, R.H.S =F1F2=11=1=F_1F_2=1 \cdot 1=1 Thus it is true for n=1n=1.

Assume it is true for n=tn=t then k=1tFk2=FtFt+1(1)\color{#4257b2}\sum_{k=1}^{t}{F_k}^2=F_tF_{t+1}\qquad \cdots (1)

Now for n=t+1n=t+1:

k=1tFk2=k=1t+1Fk2+F2t+1=(1)FtFt+1+F2t+1=Ft+1(Ft+Ft+1)=Ft+1Ft+2\begin{align*} &\sum_{k=1}^{t}{F_k}^2\\ &=\sum_{k=1}^{t+1}{F_k}^2+{F^2}_{t+1}\\ &\overset{{\color{#4257b2}(1)}}{=}F_tF_{t+1}+{F^2}_{t+1}\\ &=F_{t+1}(F_t+F_{t+1})\\ &=F_{t+1}F_{t+2} \end{align*}

Thus it is true for n=t+1n=t+1.

Therefore by mathematical induction k=1nFk2=FnFn+1\sum_{k=1}^{n}{F_k}^2=F_nF_{n+1} for every integer nNn \in \Bbb N \qquad \blacksquare

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