I am currently going through khan academy's ratio test videos/questions and have been finding it hard to calculate $a_{n+1}$ when given a specific $a_n$. The example in the video:
$a_n$ = $\frac{n^{10}}{n!}$
$a_{n+1}$ = $\frac{(n+1)^{10}}{(n+1)!}$
Seemed easy enough to understand but when there are constants involved I'm not so confident. If possible please explain fundamentally so I can apply it to additional questions. Thanks in advance.
You compute as you did in the example. But lets take the following $$ a_n = \frac{n-1}{(n+1)!} $$ and instead of using $n+1$ lets say we want to figure out $a_k$ then we would have $$ a_k = \frac{k-1}{(k+1)!} $$ now lets put back $k=n+1$ we find $$ a_{n+1} = \frac{(n+1)-1}{((n+1)+1)!} = \frac{n}{(n+2)!} $$ I hope this helps.