If $latex F_{n}$ is the nth Fibonacci Number then:
$latex \displaystyle \sum_{i=0}^{n}{i \cdot F_{2i}} = n \cdot F_{2n+1} - F_{2n}$
This identity can be easily proved using the method of induction with the basic recurrence relation of Fibonacci Numbers.
How can we find methods for constructing new identities like this one?
References:
[1]-Wikipedia - Fibonacci number
[2]-Chandra, Pravin and Weisstein, Eric W. "Fibonacci Number." From MathWorld--A Wolfram Web Resource - http://mathworld.wolfram.com/FibonacciNumber.html
