Arithmetic and Geometric Sequences and Series

Recognizing patterns in the real world

Many real situations produce numbers that follow a predictable pattern: a savings plan that adds a fixed amount each month, a population that grows by a fixed percentage each year, or machinery that loses value at a steady rate. Sequences and series give us formulas to jump straight to any term or total without listing every step.

Arithmetic sequences

An arithmetic sequence has a constant common difference $d$ between consecutive terms. The $n$th term is

$$u_n = u_1 + (n-1)d,$$

where $u_1$ is the first term. The sum of the first $n$ terms (an arithmetic series) is

$$S_n = \frac{n}{2}\left(2u_1 + (n-1)d\right) = \frac{n}{2}(u_1 + u_n).$$

Worked example. A trainee is paid \$1800 in her first month and receives a \$50 raise each month. Her salary in mont