# F_{0},\ F_{1},\ F_{2}\cdots is the Fibonacci sequence. Prove that F_{k+1}^{2}-F_{k}^{2}=F_{k-1}F_{k+2}, for all integers k\geq1

$$\displaystyle{F}_{{{0}}},\ {F}_{{{1}}},\ {F}_{{{2}}}\cdots$$ is the Fibonacci sequence.
Prove that $$\displaystyle{{F}_{{{k}+{1}}}^{{{2}}}}-{{F}_{{{k}}}^{{{2}}}}={F}_{{{k}-{1}}}{F}_{{{k}+{2}}},$$ for all integers $$\displaystyle{k}\geq{1}$$

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pierretteA

Step 1
Recall Fibonacci sequences. Let $$\displaystyle{f}_{{{n}}}$$ be the ntn term of a Fibonacci sequence,
then $$\displaystyle{f}_{{{0}}}={0},\ {f}_{{{1}}}={1},$$
$$\displaystyle{f}_{{{n}}}={f}_{{{n}-{1}}}+{f}_{{{n}-{2}}},\ {n}\geq{2},\ {n}\in{N}$$
Step 2
Now, for any integer $$\displaystyle{k}\geq{1}$$
$$\displaystyle{{F}_{{{k}+{1}}}^{{{2}}}}-{{F}_{{{k}}}^{{{2}}}}={\left({F}_{{{k}+{1}}}+{F}_{{{k}}}\right)}{\left({F}_{{{k}+{1}}}-{F}_{{{k}}}\right)}$$
$$\displaystyle={\left({F}_{{{k}+{2}}}\right)}{\left({F}_{{{k}}}+{F}_{{{k}-{1}}}-{F}_{{{k}}}\right)}$$
$$\displaystyle={\left({F}_{{{k}+{2}}}\right)}{\left({F}_{{{k}-{1}}}\right)}$$