The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13...; starting with 0 and 1, each number is the sum of the two previous numbers.

The Lucas numbers follow the same rule, but start with 2 and 1: 2, 1, 3, 4, 7, 11, 18,...

What's the sum of the first k Lucas numbers?
What's the sum of the squares of the first k Lucas numbers?

I think I found a generalized solution to any series similar to the Fibonacci series! I’m going to call it a Nikki Series just for fun =)

For any series defined as follows:
N(1) = a
N(2) = b
For k>2, N(k) = N(k-1) + N(k-2)

The following is true:
The sum of the first k Nikki numbers is N(k+2)-1 (ok, that wasn’t very interesting)
The sum of the squares of the first k Nikki numbers is
N(k)*N(k+1) + N(1)*[N(1) – N(2)]
Or
N(k)*N(k+1) + a*(a – b)

Check it out!

Posted by nikki on 2004-10-08 08:25:10