More on Sequences

A geometric sequence of the form $G = \langle a, a\cdot r, a\cdot r^2, a\cdot r^3, \dots\rangle$ is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed common ratio $r$.

The $n$th term of a geometric sequence is $a_n = a_1\cdot r^{n-1}$ (just plug in $a_1$, $r$, and $n$ to get the number.)

The sum of a geometric sequence has a formula: if the sequence is finite, like $G = \{1, 2, 4, 8, 16\}$ ($r = 2$), the formula is$$\sum_{i=1}^n a_i = a_1\cdot\frac{1 - r^n}{1 - r},\text{ ($r \neq 1$)}.$$

if the given sequence is infinite, and if $|r| < 1$, like $G = \{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots\}$ ($r = \frac{1}{2}$), the formula is$$\sum_{a \in G} a = \frac{a_1}{1 - r}.$$