# Prove $F_{1}^{2}+F_{2}^{2}+\dots+F_{n}^{2}=F_{n}F_{n+1}$ using geometric approach

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?

• Always Induction. – Grobber Aug 6 '13 at 17:01
• I said geometric approach not algebraic – user89260 Aug 6 '13 at 17:03
• Proof by induction is next-to-trivial. Why would you desire a geometric approach and how would you geometrically interpret $F_i$? – AlexR Aug 6 '13 at 17:07
• idk the homework said: in a geometry approach – user89260 Aug 6 '13 at 17:08
• – Cameron Buie Sep 29 '13 at 21:39

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.