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(I)^2 = F(k+3) -1 holds for any positive integer value of k
Edited on February 11, 2023, 4:04 am
Solution To Part A
