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?

*Solution To Part A*

We observe that:

F(1) = F(3) -1

F(1) + F(2) = 1+1 = 2 = 3-1 = F(4) -1

F(1) + F(2) + F(3) = 1+1 + 2 = 4 = F(5) -1 ...(i)

This leads us to conjecture that:

Sum(i = 1 To k) = F(k+2) -1.......(*)

Let the above conjecture be true for k = p

Then,

Sum (i = 1 To p)[F(i)] = F(p+2) - 1

Or, F(1) + F(2) + F(3)+ ......+ F(p) + F(p+1) = F(p+2) + F(p+1) -1

Or, Sum (i = 1 To p+1)[F(i)] = F(p+3) - 1

Thus the conjecture is true for k = p+1. Since, in terms of (i),

the conjecture holds for k = 1, 2, 3; it now follows that the

relationship Sum(i = 1 To k) = F(k+2) -1 holds for any positive integer value of k