## Formula for Sum of Odd Fibonacci Terms

\[F_1=F_2=1, F_{n+1}=F_{n-1}+F_n, n \ge 2\]

.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\[F_1+F_3+...+F_{2k-1}=F_{2k}\]

Adding

\[F_{2k+1}\]

to both sides.\[F_1+F_3+...+F_{2k-1}+F_{2k+1}=F_{2k}+F_{2k+1}=F_{2k+2}\]

by the definition of the Fibonacci sequence. Hence

\[P_{k+1}\]

is true and the theorem is proved.