Let {F(n)} = {1,1,2,3,5,8,13,21,34,55,···} be the Fibonacci sequence
defined by 
F(1) = F(2) = 1, F(n)= F(n-1) + F(n-2)  if n > 2. 
Show that holds for n that is greator or equal to 1.
F(2) + F(4)+....+ F(2n) = F(2n+1) - 1

Proof by induction:

It holds for n = 2 since F(2) + F(4) = 4 and F(2·2+1) - 1 = F(4+1) - 1 =
F(5) - 1 = 5 - 1 = 4 

Assume that it holds for some n = k ³ 2.

That is, 

F(2) + F(4) + ··· + F(2k) = F(2k+1) - 1

We need to show that under this assumption it also holds for n = k+1.

That is, we need to show that

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

By induction hypothesis, the left side equals to

F(2k+1)-1 + F( 2(k+1) ) = F(2k+1) + F(2k+2) - 1 which by definition

equals F(2k+3) - 1, which is what we had to prove.

QED

Edwin
AnlytcPhil@aol.com