Let $a_1=a_2=1$, $a_3=2$ and $a_n=a_{n-1}+a_{n-2}$ for $n\ge3$. Prove that $a_1 + a_2 + \ldots + a_n = a_{n+2} - 1$.
How to prove $\sum_{k=1}^{n}F_k = F_{n+2}-1$ by induction when $F_n$ is the Fibonacci sequence