Proof: Fibonacci Sequence Sums to Squares

[Euler_Euclid]

if $$a_1, a_2, a_3 ....$$ belong to the fibonacci sequence, prove that

$$a_1a_2 + a_2a_3 + ... + a_{2n-1}a_{2n} = (a_{2n})^2$$

 

[uart]

If you are familiar with mathematical induction then that's the way to go with this one. Using the recursion equation ($a_{2n} + a_{2n+1} = a_{2n+2}$ etc) should let you make the inductive step fairly easily.

BTW. Is this homework ?

 

[Euler_Euclid]

not at all!

 

[uart]

Are you familar with mathematical induction?

 

[Euler_Euclid]

Is there any other method other than this?