<meta http-equiv="refresh" content="1; url=/nojavascript/">
You are viewing an older version of this Concept. Go to the latest version.

# Finding the nth Term Given the Common Ratio and the First Term

## Sequences where ratio of any two consecutive terms is constant.

0%
Progress
Practice Finding the nth Term Given the Common Ratio and the First Term
Progress
0%
Geometric Sequences and Finding the nth Term Given the Common Ratio and the First Term

The following sequence shows the distance (in centimeters) a pendulum travels with each successive swing. Write a general rule for the geometric sequence.

80, 72, 64.8, 58.32, ...

### Guidance

A geometric sequence is a sequence in which the ratio between any two consecutive terms, $\frac{a_n}{a_{n-1}}$ , is constant. This constant value is called the common ratio . Another way to think of this is that each term is multiplied by the same value, the common ratio, to get the next term.

#### Example A

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

Is this sequence geometric? If so, what is the common difference?

Solution: If we look at each pair of successive terms and evaluate the ratios, we get $\frac{6}{2}=\frac{18}{6}=\frac{54}{18}=3$ which indicates that the sequence is geometric and that the common ratio is 3.

### More Guidance

Now let’s see if we can develop a general rule ( $n^{th}$ term) for this sequence. Since we know that each term is multiplied by 3 to get the next term, let’s rewrite each term as a product and see if there is a pattern.

$a_1 &= 2 \\a_2 &= a_1(3)=2(3)=2(3)^1 \\a_3 &= a_2(3)=2(3)(3)=2(3)^2 \\a_4 &= a_3(3)=2(3)(3)(3)=2(3)^3$

This illustrates that the general rule is $a_n=a_1(r)^{n-1}$ , where $r$ is the common ratio. This even works for the first term since $a_1=2(3)^0=2(1)=2$ .

#### Example B

Write a general rule for the geometric sequence $64, 32, 16, 8, \ldots$

Solution: From the general rule above we can see that we need to know two things: the first term and the common ratio to write the general rule. The first term is 64 and we can find the common ratio by dividing a pair of successive terms, $\frac{32}{64}=\frac{1}{2}$ . The $n^{th}$ term rule is thus $a_n=64 \left(\frac{1}{2}\right)^{n-1}$ .

#### Example C

Find the $n^{th}$ term rule for the sequence $81, 54, 36, 24, \ldots$ and hence find the $12^{th}$ term.

Solution: The first term here is 81 and the common ratio, $r$ , is $\frac{54}{81}=\frac{2}{3}$ . The $n^{th}$ term rule is $a_n=81 \left(\frac{2}{3}\right)^{n-1}$ . Now we can find the $12^{th}$ term $a_{12}=81 \left(\frac{2}{3}\right)^{12-1}=81 \left(\frac{2}{3}\right)^{11}=\frac{2048}{2187}$ . Use the graphing calculator for the last step and MATH > Frac your answer to get the fraction. We could also use the calculator and the general rule to generate terms $seq(81(2/3)^\land(x-1), x, 12, 12)$ . Reminder: the $seq ( \ )$ function can be found in the LIST ( $2^{nd}$ STAT ) Menu under OPS . Be careful to make sure that the entire exponent is enclosed in parenthesis.

Intro Problem Revisit We need to know two things, the first term and the common ratio, to write the general rule. The first term is 80 and we can find the common ratio by dividing a pair of successive terms, $\frac{72}{80}=\frac{9}{10}$ . The $n^{th}$ term rule is thus $a_n=80 \left(\frac{9}{10}\right)^{n-1}$ .

### Guided Practice

1. Identify which of the following are geometric sequences. If the sequence is geometric, find the common ratio.

a. $5, 10, 15, 20, \ldots$

b. $1, 2, 4, 8, \ldots$

c. $243, 49, 7, 1, \ldots$

2. Find the general rule and the $20^{th}$ term for the sequence $3, 6, 12, 24, \ldots$

3. Find the $n^{th}$ term rule and list terms 5 thru 11 using your calculator for the sequence $-1024, 768, -432, -324, \ldots$

### Vocabulary

Geometric Sequence
A sequence in which the ratio of any two consecutive terms is constant.
Common Ratio
The value of the constant ratio between any two consecutive terms in a geometric sequence. Also, the value by which you multiply a term in the sequence to get the next term.

### Practice

Identify which of the following sequences are arithmetic, geometric or neither.

1. $2, 4, 6, 8, \ldots$
2. $\frac{1}{2}, \frac{3}{2}, \frac{9}{2}, \frac{27}{2}, \ldots$
3. $1, 2, 4, 7, \ldots$
4. $24, -16, \frac{32}{3}, -\frac{64}{9}, \ldots$
5. $10, 5, 0, -5, \ldots$
6. $3, 4, 7, 11, \ldots$

Given the first term and common ratio, write the $n^{th}$ term rule and use the calculator to generate the first five terms in each sequence.

1. $a_1=32$ and $r=\frac{3}{2}$
2. $a_1=-81$ and $r=-\frac{1}{3}$
3. $a_1=7$ and $r=2$
4. $a_1=\frac{8}{125}$ and $r=-\frac{5}{2}$

Find the $n^{th}$ term rule for each of the following geometric sequences.

1. $162, 108, 72, \ldots$
2. $-625, -375, -225, \ldots$
3. $\frac{9}{4}, -\frac{3}{2}, 1, \ldots$
4. $3, 15, 75, \ldots$
5. $5, 10, 20, \ldots$
6. $\frac{1}{2}, -2, 8, \ldots$

Use a geometric sequence to solve the following word problems.

1. Rebecca inherited some land worth $50,000 that has increased in value by an average of 5% per year for the last 5 years. If this rate of appreciation continues, about how much will the land be worth in another 10 years? 2. A farmer buys a new tractor for$75,000. If the tractor depreciates in value by about 6% per year, how much will it be worth after 15 years?