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?

(In reply to

Solution To Part A by K Sengupta)

__Solution To Part B__

We observe that:

F(1) = F(3) - 1 = 1 = 1*1 = F(1)*F(2)

F(1)^2 + F(2)^2 = 1+1 = 2 = 1*2 = F(2)*F(3)

F(1)^2 + F(2)^2 + F(3)^2= 1+1+ 4 = 6 = F(3)*F(4)......(ii)

This leads us to conjecture that:

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

Let the above conjecture be true for k = p

Then,

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

Or, F(1)^2 + F(2)^2 + F(3)^3+ ......+ F(p)^2 + F(p+1)^2

= F(p+1) [F(p) + F(p+1)]

= F(p+1)*F(p+2)

Or, Sum (i = 1 To p+1)[F(i)^2] = F(p+1)*F(p+2)

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

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

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