13.3 Geometric Sequences

Determine if each sequence is geometric. If so, indicate the common ratio.

ⓐ $4 , 8 , 16 , 32 , 64 , 128 , \ldots$

ⓑ $−2 , 6 , −12 , 36 , −72 , 216 , \ldots$

Write the first five terms of the sequence where the first term is 3 and the common ratio is $r = −2 .$

1. ⓐ Find the thirteenth term of a sequence where the first term is 81 and the common ratio is r=1/3.
2. ⓑ Find the ninth term of the sequence 6, 18, 54, 162, 486, 1458, … Then find the general term for the sequence.

Is the sequence geometric? If so, find the common ratio.

1. ⓐ $1 , 2 , 4 , 8 , 16 , ...$
2. ⓑ $48 , 12 , 4 ,\textrm{ } 2 , ...$

Is the sequence geometric? If so, find the common ratio.

Is the sequence geometric? If so, find the common ratio.

List the first four terms of the geometric sequence with $a_{1} = 5$ and $r = –2.$

List the first five terms of the geometric sequence with $a_{1} = 18$ and $r = \frac{1}{3} .$

Write a recursive formula for the following geometric sequence.

Write a recursive formula for the following geometric sequence.

Given a geometric sequence with $a_{1} = 3$ and $a_{4} = 24 ,$ find $a_{2} .$

Given a geometric sequence with $a_{2} = 4$ and $a_{3} = 32$ , find $a_{6} .$

Write an explicit formula for the following geometric sequence.

In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.

1. ⓐWrite a formula for the student population.
2. ⓑEstimate the student population in 2020.

A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.

1. ⓐWrite a formula for the number of hits.
2. ⓑEstimate the number of hits in 5 weeks.

What is a geometric sequence?

What is the procedure for determining whether a sequence is geometric?

Describe how exponential functions and geometric sequences are similar. How are they different?

Find the common ratio for the geometric sequence.

$- 0.125 , 0.25 , - 0.5 , 1 , - 2 , ...$

Determine whether the sequence is geometric. If so, find the common ratio.

$- 6 , - 12 , - 24 , - 48 , - 96 , ...$

Determine whether the sequence is geometric. If so, find the common ratio.

$- 1 , \frac{1}{2} , - \frac{1}{4} , \frac{1}{8} , - \frac{1}{16} , ...$

Determine whether the sequence is geometric. If so, find the common ratio.

$0.8 , 4 , 20 , 100 , 500 , ...$

Write the first five terms of the geometric sequence, given the first term and common ratio.

$\begin{matrix}a_{1} = 5 , & r = \frac{1}{5}\end{matrix}$

Write the first five terms of the geometric sequence, given any two terms.

$\begin{matrix}a_{6} = 25 , & a_{8}\end{matrix} = 6.25$

Find the specified term for the geometric sequence, given the first term and common ratio.

The first term is 16 and the common ratio is $- \frac{1}{3} .$ Find the 4^th term.

Find the specified term for the geometric sequence, given the first four terms.

$a_{n} = \left\{- 2 , \frac{2}{3} , - \frac{2}{9} , \frac{2}{27} , ...\right\} .$ Find $a_{7} .$

Write the first five terms of the geometric sequence.

$\begin{matrix}a_{1} = 7 , & a_{n} = 0.2 a_{n - 1}\end{matrix}$

Write a recursive formula for each geometric sequence.

$a_{n} = \left\{- 32 , - 16 , - 8 , - 4 , ...\right\}$

Write a recursive formula for each geometric sequence.

$a_{n} = \left\{10 , - 3 , 0.9 , - 0.27 , ...\right\}$

Write a recursive formula for each geometric sequence.

$a_{n} = \left\{\frac{3}{5} , \frac{1}{10} , \frac{1}{60} , \frac{1}{360} , ...\right\}$

Write a recursive formula for each geometric sequence.

$a_{n} = \left\{\frac{1}{512} , - \frac{1}{128} , \frac{1}{32} , - \frac{1}{8} , ...\right\}$

Write the first five terms of the geometric sequence.

$a_{n} = 12 \cdot \left(- \frac{1}{2}\right)^{n - 1}$

Write an explicit formula for each geometric sequence.

$a_{n} = \left\{1 , 3 , 9 , 27 , ...\right\}$

Write an explicit formula for each geometric sequence.

$a_{n} = \left\{0.8 , - 4 , 20 , - 100 , ...\right\}$

Write an explicit formula for each geometric sequence.

$a_{n} = \left\{- 1 , - \frac{4}{5} , - \frac{16}{25} , - \frac{64}{125} , ...\right\}$

Write an explicit formula for each geometric sequence.

$a_{n} = \left\{3 , - 1 , \frac{1}{3} , - \frac{1}{9} , ...\right\}$

Find the specified term for the geometric sequence given.

Let $a_{n} = - \left(- \frac{1}{3}\right)^{n - 1} .$ Find $a_{12} .$

Find the number of terms in the given finite geometric sequence.

$a_{n} = \left\{2 , 1 , \frac{1}{2} , ... , \frac{1}{1024}\right\}$

Use recursive formulas to give two examples of geometric sequences whose 3^rd terms are $200.$

Find the 5^th term of the geometric sequence $\left\{b , 4 b , 16 b , ...\right\} .$

At which term does the sequence $\left\{10 , 12 , 14.4 , 17.28 , ...\right\}$ exceed $100 ?$

For which term does the geometric sequence $a_{_{n}} = - 36 \left(\frac{2}{3}\right)^{n - 1}$ first have a non-integer value?

Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8^th term.