6
$\begingroup$

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?

$\endgroup$
  • 2
    $\begingroup$ Always Induction. $\endgroup$ – Grobber Aug 6 '13 at 17:01
  • 1
    $\begingroup$ I said geometric approach not algebraic $\endgroup$ – user89260 Aug 6 '13 at 17:03
  • $\begingroup$ Proof by induction is next-to-trivial. Why would you desire a geometric approach and how would you geometrically interpret $F_i$? $\endgroup$ – AlexR Aug 6 '13 at 17:07
  • $\begingroup$ idk the homework said: in a geometry approach $\endgroup$ – user89260 Aug 6 '13 at 17:08
  • $\begingroup$ possible duplicate of For the Fibonacci numbers, show for all $n$: $F\_1^2+F\_2^2+\dots+F\_n^2=F\_nF\_{n+1}$ $\endgroup$ – Cameron Buie Sep 29 '13 at 21:39
15
$\begingroup$

enter image description here

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.

$\endgroup$
  • $\begingroup$ sounds good appreciatted! $\endgroup$ – user89260 Aug 6 '13 at 17:14
  • $\begingroup$ That is too cool! $\endgroup$ – Robert Lewis Sep 5 '13 at 19:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.