The formula for the nth term is n(n+1)/2, where n=0, 1, 2, etc.
To derive this formula for the sum of the natural numbers up to n, we use the fact that the average of the first and last numbers (0 and n), the average of the second and penultimate numbers (1 and n-1), etc. is constant, and we have n+1 of these, hence the formula.