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Fibonacci sums (Posted on 2004-07-15) Difficulty: 3 of 5
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?

See The Solution Submitted by Federico Kereki    
Rating: 3.7143 (7 votes)

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Solution Solution To Part A | Comment 12 of 13 |

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


  Posted by K Sengupta on 2007-05-31 06:12:27
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