Introduction to Geometric Sequences

You already know that arithmetic sequences increase or decrease by a fixed common difference $d$ . Geometric sequences work differently — instead of adding the same value each time, you multiply by a fixed value called the common ratio.

Geometric Sequence

A sequence in which each term is obtained by multiplying the previous term by a fixed constant called the common ratio, $r$ . The sequence is defined by its first term $u_{1}$ and the value of $r$ .

The general form of a geometric sequence looks like this:
$u_{1}, u_{1} r, u_{1} r^{2}, u_{1} r^{3}, \dots, u_{1} r^{n - 1}$

Notice that the power of $r$ is always one less than the position of the term — the 1st term has $r^{0} = 1$ , the 2nd has $r^{1}$ , the 3rd has $r^{2}$ , and so on.

Example

Consider the sequence $2, 6, 18, 54, \dots$

First term: $u_{1} = 2$
Common ratio: $r = \frac{6}{2} = 3$

The sequence can be written as $2, 2 (3), 2 (3^{2}), 2 (3^{3}), \dots$ , confirming each term is the previous one multiplied by 3.

Free preview

SL 1.3—Geometric sequences and series

Introduction to Geometric Sequences

7 more sections in this topic