2016/001) How to find the nth term of a sequence whose consecutive terms difference' is in A.P?

for example $1,3,6,10,15\cdots$

Because difference is AP let us find the differences  $2,3,4,5$
$t_{n+1} - t_{n} = n$
so $t_n$ should be order 2 polynomial
say $t_n = an^2 + bn + c$
hence $t_{n+1}-t_n=a((n+1)^2-n^2) + b=2an +a +b = n $
so $2a = 1$ and $a+b = 1 => a = b = \frac{1}{2}$
putting $n = 1$ we get $c = 0$
so $t_n = \frac{1}{2}(n^2 + n) = \frac{1}{2}n(n+1)$