Say you have +1 on first iteration, +2 on second, and so on until N, and you want to know the total. That's easily calculate using (N * (N + 1) ) / 2. What's that equation or technique called?
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The term you're looking for is a triangular number.
The "technique", so to speak, is called summation, usually denoted with a $\Sigma$, like so (I should note that the formula you've listed is incorrect): $$\sum_{k=1}^{n}{k}=\frac{n(n+1)}{2}$$
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fixed it thanks – zipquincy Jun 11 '14 at 13:38
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Depending on which technique you are talking about, one technique was discovered by Gauss when he was about 8 years old.
His teacher supposedly posed the question of finding the sum of the first $100$ positive integers to keep the class busy during the long periods, and it turns out Gauss found a way to do it quickly. You could call the technique Gauss' Trick, Gauss' Method, Gaussian method.
The numbers are called Triangular Numbers. The $N$th triangular number, which corresponds to the sum of the first $N$ positive integers, is given by
$$T_n = \frac{n(n+1)}{2}$$
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