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

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.