Geometric Sequence

A geometric sequence has a constant ratio between consecutive terms called the common ratio and is represented by the variable r.

Let's see if you can determine if the following sequences are geometric.

1.) 4, 12, 16, 20, 22 ...

Solution: Find the ratio between the consecutive terms.

$LaTeX: \frac{12}{4}=3$

$LaTeX: \frac{16}{12}=\frac{4}{3}$

$LaTeX: \frac{20}{16}=\frac{5}{4}$

$LaTeX: \frac{22}{20}=\frac{11}{10}$

This is not a geometric sequence because the ratios are different.

2.) 256, 64, 16, 4, 1, ...

Solution: Find the ration between the consecutive terms.

$LaTeX: \frac{64}{256}=\frac{1}{4}$

$LaTeX: \frac{16}{64}=\frac{1}{4}$

$LaTeX: \frac{4}{16}=\frac{1}{4}$

$LaTeX: \frac{1}{4}$

Yes, this is a geometric sequence because there is a common ratio.

Recursive Formula

If we are given the first term of the sequence and the common ratio, we can find any term in the sequence by multiplying the previous term with the r, the common ratio.

Example 1: Find the next three terms of the sequence.

16, 24, 36, __, __, __

Solution: Divide the two consecutive terms to find the common ratio:

$LaTeX: \frac{24}{16}=\frac{3}{2}$ ; $LaTeX: \frac{36}{24}=\frac{3}{2}$ Common ratio = $LaTeX: \frac{3}{2}$ so the next three numbers are: 54, 81, 121.5

Example 2: Find the first five terms of the geometric sequence.

LaTeX: a_1=2,r=-3

Solution: Use the recursive formula.

LaTeX: a_1=2

$LaTeX: a_2=2\cdot(-3)=-6$

$LaTeX: a_3=-6\cdot(-3)=18$

$LaTeX: a_4=18\cdot(-3)=-54$

$LaTeX: a_5=-54\cdot(-3)=162$

Explicit Formula

The explicit formula of a geometric sequence allows us to find any term in an arithmetic sequence if we know the first term and the common ratio.

Example: Find the indicated term of each geometric sequence.

1. Find the 8th term given LaTeX: a_1=-3,r=2

Solution: Since we are looking for the 8th term, n=8.

$LaTeX: a_8=-3\cdot(2)^{8-1}$

$LaTeX: a_8=-3\cdot(2)^7=-3\cdot128=-384$

The 8th term is -384.

2. Find the 10th term of the sequence given

LaTeX: a_3=99 and r= -3

Solution: First find the first tern, LaTeX: a_1 .

$LaTeX: a_3=a_1\cdot(-3)^{3-1}$

$LaTeX: 99=a_1\cdot(-3)^2$

$LaTeX: 99=a_1\cdot9$

LaTeX: a_1=11

Now find the 15th term.

$LaTeX: a_{10}=11\cdot(-3)^{10-1}$

$LaTeX: a_{10}=11\cdot(-3)^9$

$LaTeX: a_{10}=11\cdot(-19,683)$

$LaTeX: a_{10}=-216,513$

Take notes as you watch the next three videos. You will learn more about geometric sequences and how to use your calculator to help you solve.

You try: Complete the practice problems below to check your understanding.