Suppose f n is the nth Fibonacci number. Prove that
f1^2*f2^2 +....Fn^2 = fn*fn+1 where n is a positve integer.
Any ideas????
Let's think geometrically.
By definition $\displaystyle f_{n+1}=f_n+f_{n-1}$
Let us draw a square of side-length $\displaystyle f_1=1$ first, then put another one but with side-length $\displaystyle f_2=1$ beisde the other one. So that they share a side. Now we can put a square of side $\displaystyle f_3=2$ right next to the other ones so that a border is shared and so on...
For example see here
Note now that, the area of the big rectangle is equal to the sum of the areas of the small squares inside. So, if the sides of the rectangle are $\displaystyle f_n$ and $\displaystyle f_{n+1}$, it follows that $\displaystyle f_{n+1}\cdot{f_n}=f_n^2+f_{n-1}^2+...+f_1^2$
You may also try using induction