Arithmetic Sequences and Series Worked Examples Cheat Sheet

Key Facts

• An arithmetic sequence has a constant common difference $d$, so $d = a_n - a_{n-1}$ for consecutive terms.
• The explicit formula for the $n$th term is $a_n = a_1 + (n - 1)d$.
• The recursive formula is $a_1$ given and $a_n = a_{n-1} + d$ for $n \ge 2$.
• To find the term number, rearrange $a_n = a_1 + (n - 1)d$ and solve for $n$.
• The sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$ when the first and last terms are known.
• The sum formula $S_n = \frac{n}{2}(2a_1 + (n - 1)d)$ is useful when $a_1$, $d$, and $n$ are known.
• For the sequence $7, 11, 15, 19, \ldots$, the common difference is $d = 4$ and the explicit formula is $a_n = 7 + 4(n - 1)$.
• For an arithmetic series, the average of the first and last terms is $\frac{a_1 + a_n}{2}$, so $S_n = n \cdot \frac{a_1 + a_n}{2}$.

Vocabulary

Arithmetic sequence

An arithmetic sequence is a list of numbers where each term is found by adding the same common difference $d$.

Common difference

The common difference $d$ is the constant amount added to one term to get the next term.

Explicit formula

An explicit formula gives the value of $a_n$ directly using the term number $n$.

Recursive formula

A recursive formula defines each term using the previous term, such as $a_n = a_{n-1} + d$.

Arithmetic series

An arithmetic series is the sum of the terms of an arithmetic sequence.

Partial sum

A partial sum $S_n$ is the sum of the first $n$ terms of a sequence.