how to find $a_{50}$ from a recursive term Given $a_{n+1}=a_n+2n+3,a_1=3$
How can I find $a_{50}$?
I can compute $a_2,a_3,...,a_{50}$
But it's a long way. Is there any smart technique to compute?
Thanks.

Do a "list":
$$\begin{align}&a_1=3\\&a_2=a_1+2+3\\&\ldots\\&a_n=a_{n-1}+2(n-1)+3\\&a_{n+1}=a_n+2n+3\end{align}$$
Now sum up both sides
$$\sum_{k=1}^{n+1}a_k=\sum_{k=1}^na_k+3(n+1)+2\sum_{k=1}^nk\implies$$
$$a_{n+1}=3(n+1)+2\sum_{k=1}^nk=3(n+1)+n(n+1)=(n+3)(n+1)$$
