How to find a recurrence relation for the following sequence

Question

I have to find a recurrence relation that generates the sum of the first $n$ cubes, that is $s_n = 1 + 8 + 27 + \dots + n^3$ considering that $n=1,2,3,\dots$

I also have to find a recurrence relation for $s_n = n!$ and find the initial condition.

EDIT: My answer for the first one is

$$s_n=s_{n-1}+3n^2-3n+1$$

and the second one as:

$$s_n=ns_{n-1}$$

With both initial terms as:

$$s_1=1$$

Would this be correct?

Answer 1

For the first, how about $s_0=1$ and

$$s_n -s_{n-1} = n^3$$