Recurrence relations help please?

Question

How do I solve this recurrence relation?

$$ a_k = a_{k-1} + k $$ when $a_0 = 2$.

Answer 1

hint: $a_k = (a_k-a_{k-1})+(a_{k-1}-a_{k-2})+\cdots + (a_2-a_1) +(a_1-a_0)+a_0 = k+(k-1)+(k-2)+\cdots + 2+1+2$

Answer 2

Note that $$ a_k - a_{k-1} = k, $$ so $$ a_{n}-a_0 = \sum_{k=1}^{n} (a_k-a_{k-1}) = \sum_{k=1}^n k = \frac{1}{2}n(n+1). $$