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