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This is very simple, but I am trying to simplify down Fibonacci sequences when doing induction problems, and I am very confused by this. For example, I was trying to simplify $F_n * F_{n+1} + F^2_{n+1})$ down to $F_{n+1} * F_{n+2}$

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    what is $k{}{}{}{}$? – 2017-02-26
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    You probably want to use `F_{n+1}`, not `F_(n+1)`. – 2017-02-26
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    Thanks I was trying to figure that out. And that k was supposed to be an n. – 2017-02-26

2 Answers 2

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You just have to use the definition of the Fibonacci sequence:

$$F_{n+2}=F_{n}+F_{n+1}$$

Factoring $F_{n+1}$ we have,

$$F_{n+1}(F_{n}+F_{n+1})$$

Now it should be clear.

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$F_nF_{n+1}+F_{n+1}^2=(F_n+F_{n+1})F_{n+1}=F_{n+2}F_{n+1}$