Arithmetic and Geometric Sequences

Constant difference $d$.

$a_n = a_1 + (n-1)d$

$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(2a_1 + (n-1)d)$

Example

$3, 7, 11, 15, ...$. $d = 4$. $a_{20} = 3 + 19(4) = 79$.

Constant ratio $r$.

$a_n = a_1 \cdot r^{n-1}$

$S_n = \frac{a_1(1 - r^n)}{1 - r}$

Sum to infinity ($|r| < 1$): $S_\infty = \frac{a_1}{1-r}$.

Example

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

• Identify the type first: constant difference → arithmetic, constant ratio → geometric.
• The ACT may give you two terms and ask for another — set up equations to find $a_1$ and $d$ (or $r$).