Answer
See the explanation
Work Step by Step
Let $P(n)$ denote the statement $F_1+F_3+\ldots +F_{2n-1}=F_{2n}$.
Step 1.
$P(1)$ is the statement that $F_1=F_2$, or $1=1$, which is true.
Step 2.
Assume that $P(k)$ is true. Thus our induction hypothesis is $F_1+F_3+\ldots+F_{2k-1}=F_{2k}$.
We want to show $P(k+1)$ is true.
$F_1+F_3+\ldots+F_{2k-1}+F_{2(k+1)-1}=F_{2k}+F_{2(k+1)-1}$ (By Induction hypothesis)
$=F_{2k}+F_{2k+1}$ (Use the formula $F_n=F_{n-1}+F_{n-2}$ for $n=2k+2$)
$=F_{2k+2}$
$=F_{2(k+1)}$
Thus $P(k+1)$ follows from $P(k)$, and this completes the induction step.
