recurrence relation for the sum of consecutive numbers

Hi guys...I'm just starting to study recurrence relations and i
thought they could come in handy to proof the formula:

1+2+3+...+n= 1/2 (n^2 + n)

from which

m + (m+1) +( (m+1) + (m+2) ) +....+n = 1/2 (n^2 + n - m^2 + m)

( so that I can calculate the sum of consecutive numbers starting from
a generic number m different from 1)

I can easily prove these two formulas geometrically but I would like
to prove them through recurrence relations.

I set:

X (n) = X (n-1) +n

with X (0) = 0
X(1) = 1

But I am not able to solve this equation perhaps because instead of a
specific number I put n which is also the index of X.
I would be glad if someone could tell me either how to solve this
equation or how to change it!

thanks in advance for your help
.