Sequence Summation

Geometric Series: $a, a r, a r^{2}, \dots, a r^{n}$

S_{n} = a \cdot \frac{1 - r ^{n}}{1 - r}

If it converges, the infinite sum:

S_{\infty} = \frac{a}{1 - r}

Arithmetic Sequence: $a, a + d, a + 2 d, \dots a + n d$

S_{n} = \frac{n}{2} \cdot (2 a + (n - 1) d)

Properties of Summations

(i = 0 \sum n a_{i})^{2} = i = 0 \sum n a_{i}^{2} + i \neq = j \sum n a_{i} a_{j}