Summation Properties

You can manipulate summations algebraically. Here are the key properties:

$\bullet$ Factor out constants: $\sum_{i=1}^{n} c \cdot a_i = c \sum_{i=1}^{n} a_i$

$\bullet$ Split sums: $\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$

$\bullet$ Change of index: $\sum_{i=1}^{n} a_i = \sum_{i=0}^{n-1} a_{i+1}$

These rules let you transform summations to match known formulas. If your code computes $\sum_{i=1}^{n} (3i + 2)$, split it: $3\sum_{i=1}^{n} i + \sum_{i=1}^{n} 2 = 3 \times \frac{n(n+1)}{2} + 2n$.