Algebra and Trigonometry, 5th Edition

11.3 Geometric Sequences and Series

• Identify the common ratio of a geometric sequence, and find a given term and the sum of the first n terms.

• Find the sum of an infinite geometric series, if it exists.

A sequence in which each term after the first is found by multiplying the preceding term by the same number is a geometric sequence.

Geometric Sequences

Consider the sequence:

$$2,6,18,54,162,\dots .$$

Note that multiplying each term by 3 produces the next term. We call the number 3 the common ratio because it can be found by dividing any term by the preceding term. A geometric sequence is also called a geometric progression.

Geometric Sequence

A sequence is geometric if there is a number r, called the common ratio, such that

$$\frac{a_{n+1}}{a_n}=r,\text{or}{a}_n$$