Prove that $F_{n}^{2}+F_{n+1}^{2}=F_{2n+1}$

Question

Prove that $F_{n}^{2}+F_{n+1}^{2}=F_{2n+1}$

This identity holds for $n>=1$

Instead of using induction, how do I prove it in a geometry approach?

Answer 1

First create a rectangle with length $F_{2n+1}$ and height $F_1 = 1$. Then cut this up into two rectangles with sizes $F_{2n} \times F_2$ (i.e. $F_{2n} \times 1$) and $F_{2n-1} \times F_1$ (i.e. $F_{2n-1} \times 1$). Put the latter under the former.

Then cut the resulting figure into rectangles with sizes $F_{2n-1} \times F_3$ and $F_{2n-2} \times F_2$. Again, put the latter under the former.

Now cut the resulting figure into rectangles $F_{2n-2} \times F_4$ and $F_{2n-3} \times F_3$ and put the latter under the former.

Eventually, after $n-1$ steps you will get a figure built up from two squares with sizes $F_{n+1}^2$ and $F_n^2$.

See below figure for an example with $n=6$, i.e. $F_{2n+1} = F_{13} = 233$. The first line is the $F_{13} \times F_1$ rectangle. After $n-1=5$ steps you will reach a figure which can be divided into $F_7^2$ and $F_6^2$ as desired.