The Fibonacci series 0, 1, 1, 2, 3, 5, 8, 13, in which each number is the sum of the two previous, is defined as F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2) for n>1.

What is the sum of F(0)+F(1)+F(2)+...+F(k)?
What is the sum of F(0)^2+F(1)^2+F(2)^2+...+F(k)^2?

From simple observation, the sum of F(0) thru F(k) = F(k+2) - 1

The sum of F(0)^2 thru F(k)^2 seems to be F(k+1)^2 - F(k), plus 1 if k is even and minus 1 if k is odd.  My bet is there is a more elegant solution to this part.

Posted by Bryan on 2004-07-15 14:53:26