## Formula for Sum of Odd Fibonacci Terms

The Fibonacci sequence is defined by
$F_1=F_2=1, F_{n+1}=F_{n-1}+F_n, n \ge 2$
.
Theorem
$F_1+F_3+...+F_{2n-1}=F_{2n}$
.   (1)
Proof is by induction. Let
$P(k)$
be the statement that (1) is true for
$n=k$
. Then
$P(1)$
is true since
$F_1=F_2$
.
Suppose that
$P(k)$
is true. Then
$F_1+F_3+...+F_{2k-1}=F_{2k}$

$F_{2k+1}$
$F_1+F_3+...+F_{2k-1}+F_{2k+1}=F_{2k}+F_{2k+1}=F_{2k+2}$
$P_{k+1}$