Arithmetic Sequences

What is a Sequence?

A sequence is an ordered list of numbers.

Example: $2, 5, 8, 11, 14, ...$

Each number is called a term.

Arithmetic Sequence

An arithmetic sequence has a common difference ($d$) between consecutive terms.

Example: $3, 7, 11, 15, 19, ...$

• Common difference: $d = 4$

Finding the Common Difference

$d = a_2 - a_1 = a_3 - a_2 = ...$

Subtract any term from the next term.

Explicit Formula

To find the $n$th term: $a_n = a_1 + (n - 1)d$

where:

• $a_n$ = nth term
• $a_1$ = first term
• $d$ = common difference
• $n$ = term number

Recursive Formula

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

Each term equals the previous term plus $d$.