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

preprekomW 2021-08-10 Answered
\(\displaystyle{F}_{{{0}}},\ {F}_{{{1}}},\ {F}_{{{2}}}\cdots\) 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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Expert Answer

pierretteA
Answered 2021-08-11 Author has 24744 answers

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
Prove what is asked.
Now, for any integer \(\displaystyle{k}\geq{1}\)
\(\displaystyle{{F}_{{{k}+{1}}}^{{{2}}}}-{{F}_{{{k}}}^{{{2}}}}={\left({F}_{{{k}+{1}}}+{F}_{{{k}}}\right)}{\left({F}_{{{k}+{1}}}-{F}_{{{k}}}\right)}\)
\(\displaystyle={\left({F}_{{{k}+{2}}}\right)}{\left({F}_{{{k}}}+{F}_{{{k}-{1}}}-{F}_{{{k}}}\right)}\)
\(\displaystyle={\left({F}_{{{k}+{2}}}\right)}{\left({F}_{{{k}-{1}}}\right)}\)

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