Sequences and Series

Definitions

• Sequence: An ordered list of numbers following a pattern
• Series: The sum of terms in a sequence
• Term: Each number in a sequence, denoted $a_n$

Arithmetic Sequences

A sequence where the difference between consecutive terms is constant.

Common Difference

$d = a_{n+1} - a_n$

nth Term Formula

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

Alternative Form

$a_n = a_m + (n - m)d$

Sum of First n Terms (Arithmetic Series)

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

or

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

Geometric Sequences

A sequence where the ratio between consecutive terms is constant.

Common Ratio

$r = \frac{a_{n+1}}{a_n}$

nth Term Formula

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

Sum of First n Terms (Geometric Series)

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

or

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

Infinite Geometric Series ($|r| < 1$)

$S_\infty = \frac{a_1}{1 - r}$

Note: Only converges when $|r| < 1$

Summation Notation

Sigma Notation

$\sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \cdots + a_n$

Properties

$\sum(a_i + b_i) = \sum a_i + \sum b_i$

$\sum(c \cdot a_i) = c \cdot \sum a_i$

$\sum c = n \cdot c \quad \text{(summing constant } n \text{ times)}$

Common Summation Formulas

Sum	Formula
$\sum_{i=1}^{n} 1$	$n$
$\sum_{i=1}^{n} i$	$\frac{n(n + 1)}{2}$
$\sum_{i=1}^{n} i^2$	$\frac{n(n + 1)(2n + 1)}{6}$
$\sum_{i=1}^{n} i^3$	$\left[\frac{n(n + 1)}{2}\right]^2$
$\sum_{i=1}^{n} r^{i-1}$	$\frac{1 - r^n}{1 - r}$

Recursive Formulas

A formula that defines each term using previous terms.

Arithmetic

$a_n = a_{n-1} + d, \quad \text{given } a_1$

Geometric

$a_n = r \cdot a_{n-1}, \quad \text{given } a_1$

Fibonacci Sequence

$F_n = F_{n-1} + F_{n-2}$

$F_1 = 1, \quad F_2 = 1$

$1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$

Special Sequences

Triangular Numbers

$T_n = 1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}$

$1, 3, 6, 10, 15, 21, \ldots$

Square Numbers

$n^2 = 1, 4, 9, 16, 25, 36, \ldots$

Cube Numbers

$n^3 = 1, 8, 27, 64, 125, \ldots$

Arithmetic vs Geometric Comparison

Property	Arithmetic	Geometric
Pattern	Add constant $d$	Multiply by $r$
nth term	$a_1 + (n-1)d$	$a_1 \cdot r^{n-1}$
Sum formula	$\frac{n}{2}(a_1 + a_n)$	$\frac{a_1(1 - r^n)}{1 - r}$
Infinite sum	Diverges	$\frac{a_1}{1-r}$ if $