How to use undetermined coefficient in Difference Equations (Recurrence Relation)

Question

I have $a_n-a_{n-1}=n$ with some initial conditions $a_0=1, a_2=2, a_3=4, a_4=7$ since there exist $n$, it's inhomogeous equation isn't it?

I know from ODE i can bring the homogenous solution and differentiate it and do a substitution into the original equation to find the particular solution.

But how about this difference equation?

Answer 1

Here is a solution:

$a_n-a_0=a_n-a_{n-1}+a_{n-1}-a_{n-2}+a_{n-2}-a_{n-3}+...+a_1-a_0$

$=n+(n-1)+(n-2)+...+(n-(n-1))$

$=n\times n - \frac {(n-1)n}2=\frac{n^2}2+\frac n2,$

so $a_n=\frac{n^2}2+\frac n2+1$.