When you're given a sequence, look at the differences between consecutive terms:
terms: 1 3 6 10 15 21
differences: 2 3 4 5 6
The differences grow by 1 each time, so term n is 1 + 2 + … + n: the triangular numbers, n(n+1)/2.
General tactic: if the first differences are constant, the formula is linear. If the second differences are constant, it's quadratic. Then solve for the coefficients.