Arithmetic and Geometric Sequences

An arithmetic sequence has a constant difference $d$ between terms:
$$a_n = a_1 + (n-1)d$$

Example: $3, 7, 11, 15, ...$ has $a_1 = 3$, $d = 4$.
$a_{10} = 3 + 9(4) = 39$.

Sum of first $n$ terms: $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$

A geometric sequence has a constant ratio $r$ between terms:
$$a_n = a_1 \cdot r^{n-1}$$

Example: $2, 6, 18, 54, ...$ has $a_1 = 2$, $r = 3$.
$a_5 = 2 \cdot 3^4 = 162$.

Sum of first $n$ terms: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ (for $r \neq 1$)