TITLE: Prove $F_{1}^{2}+F_{2}^{2}+\dots+F_{n}^{2}=F_{n}F_{n+1}$ using geometric approach
QUESTION [5 upvotes]: Prove the identity:  $F_{1}^{2}+F_{2}^{2}+\dots+F_{n}^{2}=F_{n}F_{n+1}$, where $F_i$ denotes a Fibonacci number.

How can I prove it using a geometric approach?

REPLY [16 votes]: The horizontal side length is $F_{n+1}$, in this case $21 + 13=34$. The vertical side is $F_n$, in this case $13 + 8 = 21$. However, the area can also be defined as the sum of the smaller squares.