Задача 12.

(the Fibonacci numbers) Let $u(n)$ be a sequence of numbers such that $u(0)=0, u(1)=1, u(n)=u(n-1) + u(n-2)$. 1) Show that $u(1) + \ldots + u(n) = u(n+2) - 1$. 2) Show that $(u(1))^2 + \ldots + (u(n))^2 = u(n)\cdot{u(n+1)}$. 3) (Binet's formula) What do you think is the relation between numbers $u(n)$ and $\delta(n)= \frac{1}{\sqrt{5}}(\frac{1+\sqrt{5}}{2})^n-\frac{1}{\sqrt{5}} (\frac{1-\sqrt{5}}{2})^n$?