Compare the summation below:
$$\begin{align}
\smash[b]{\sum\_{i=1}^n F\_{2i-1}}&=F\_1+F\_3+F\_5+\cdots+F\_{2n-1}\\
&=1+2+5+\cdots+F\_{2n-1}\\
&=F\_{2n}\\
\end{align}
$$
with this one:
$$\begin{align}
\smash[b]{\sum\_{i=1}^n F\_{2i}}&=F\_2+F\_4+F\_6+\cdots+F\_{2n}\\
&=1+3+8+\cdots+F\_{2n}\\
&=F\_{2n+1}-1\\
\end{align}$$
When I first discovered these patterns I was amazed. Naively I had thought that an every-other-number sum of Fibonacci numbers would be the same pattern whether the parity of their indices was odd or even, but I was wrong! Why is the above true, where the summation of odd-indexed Fibonacci numbers is another Fibonacci number, but the even-indexed sum is a Fibonacci number minus 1?