Arithmetic and geometric sequences

The big idea

A sequence is a list of numbers that follows a rule. Each number is a term, and we write the $n$th term as $u_n$. Two rules appear again and again:

• Keep adding the same amount — an arithmetic sequence.
• Keep multiplying by the same amount — a geometric sequence.

Adding up the terms of a sequence gives a series.

Explained simply

Think of saving money. If you put in 20 more every week, your savings grow arithmetically: steady steps. If your money earns interest and grows by 5% every year, it grows geometrically: each step is bigger than the last.

Arithmetic. The fixed amount added is the common difference $d$. To reach term you start at and take steps of size :