Vtyjkyuk

Find the general term of $x_n+1 = fracx_n + 1n + 1$ where $x_1 = 0$

I've tried finding the general term by introducing a generating function $G(z)$ and then solving for $G(z)$ with no luck.

I've also tried to express $x_n$ and get rid of the constant doing the following:

$$
x_n+1 = fracx_n + 1n+1\
x_n = fracx_n-1 + 1n
$$

Subtract one from another:

$$
(n+1)cdot x_n+1 - ncdot x_n = x_n - x_n-1
$$

So:
$$
(n+1)cdot (x_n+1 - x_n) = -x_n-1
$$

Not sure how to proceed from here. I've seen how recurrence relations are solve with characteristic equations. Should i introduce some characteristic polynomial and solve it?

Basically i'm more interested in a common approach than in solution (yet an example would be nice) since i am going to solve lots of similar problems after this one.

Hint. Let $z_n=n!x_n$ then $z_0=0$ and
$$z_n+1=(n+1)!x_n+1=n!(n+1)x_n+1=n!(x_n+1)=z_n+n!\=
z_n-1+(n-1)!+n!=sum_k=1^n k!$$

Another way to find the same answer : let $x_n=fraca_nb_n$ (with $a_n$ and $b_n$ integers), then
$$x_n+1=frac1+fraca_nb_nn+1=fraca_n+b_n(n+1)b_n$$
so that
$$a_n+1=a_n+b_ntext and b_n+1=(n+1)b_n$$
with initial conditions $a_1=0$ and $b_1=1$.
Solving this, you find $b_n=n!$, and $a_n=sum_k=1^n-1k!$, so
$$x_n=frac1!+2!+dots+(n-1)!n!$$