Proving a Recursion Using Induction I am trying to prove the following recursion.
$$a(n) = \left\{\begin{matrix}
n(a(n-1)+1) & \text{if } n \geq 1\\ 
0 & \text{if } n = 0
\end{matrix}\right.$$
is the series definition of $a(n)$. using this, I need to prove that
$$ a(n) = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)!}\bigg)$$
for $n \geq 1$ by induction on $n$.
I've found that the $n$ equals, for the first 5 terms, $2,5,16,65,326$. I think now I need to find a formula that describes these terms, and therefore $a(n)$. The problem is, I don't know where to start. Can anyone give me a hand?

Your induction step is:
\begin{align}
n \cdot a_{n-1} + n & = n(n-1)!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)!}\bigg) + 1 \\
& = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)!}\bigg) + n! \cdot \frac{1}{n!} \\
& = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)}!  + \frac{1}{n!}\bigg) \\
& = a_n
\end{align}
The base is easy. Then you're done. Read up on how induction works.
