Define a Geometric Sequence by the nth Term

4.18.4 Define a Geometric Sequence by the nth Term

4.18.4 • Define a Geometric Sequence by the nth Term

Activity

In 2015, the number of wildcats in a national park was 284. It was estimated that the wildcat population increased by 4% each year.

1. Instead of writing a recursive definition, a researcher for the park writes $W_{n} = 284 (1.04)^{n}$ , where $W$ is the projected wildcat population $n$ years after 2015. Explain where the different factors in her expression came from.

Compare your answer:

Your answer may vary, but here are some samples.

The 284 is the starting population of the wildcats in 2015.
The 1.04 represents that the population grows 4% or 0.04 each year.
The 1.04 comes from 100% of the population + 4% growth = 104% or 1.04.
The $n$ represents the number of years since 2015. So, in 2016, $n = 1$ and in 2017, $n = 2$ .

Work with a partner to discuss and create a table of values to estimate the projected wildcat population over the next 7 years.

2. Complete the table when $n = - 2$ on your paper and check your answers.

Year	$n$	Wildcat Population
2015	0	$W_{n} = 284 (1.04)^{0} = 284$
2016	1	$W_{n} = 284 (1.04)^{1} = 295$
2017	2	$W_{n} = 284 (1.04)^{2} =$
2018	3
2019	4
2020	5
2021	6
2022	7

Compare your answer:

Year	$n$	Wildcat Population
2015	0	284
2016	1	295
2017	2	307
2018	3	319
2019	4	332
2020	5	346
2021	6	359
2022	7	374

3. Use the graphing tool or technology outside the course. Graph the data that represents the population projections for this scenario.

Access multimedia content

Compare your answers:

A scatter plot with blue circles showing a positive linear trend. The x-axis ranges from negative 2 to 8, and the y-axis ranges from negative 100 to 550. Data points increase gradually from left to right. Data points are from the provided table.

4. Calculate and enter the year when the wildcat population will exceed 500 members.

Compare your answer:

The wildcat population is expected to exceed 500 members in 2030 when they are projected to have 511 wildcats.

5. Is the data that represents this scenario an arithmetic or geometric sequence? Explain your reasoning.

Compare your answer:

Your answer may vary, but here are some samples:

This data represents a geometric sequence because the equation has a common ratio of 1.04.
This data represents a geometric sequence because the ratio between consecutive terms in the table is 1.04.
This data represents a geometric sequence because the graph does not depict a line, it is curved a little and does not have a constant slope.

Remember that a recursive formula defines terms using one or more of the previous terms. If you need to calculate the $100^{t h}$ term or the $500^{t h}$ term in a sequence, the recursive formula becomes difficult to use if you do not know the values of the terms near $n = 100$ or $n = 500$ .

Different definitions can often create the same sequence but are more generalizable. These are called general rules or explicit rules. These formulas can be used to find any term in the sequence and may also be referred to as the rule to find the nth term.

Geometric Sequence Formulas

Recursive Formula

$a_{n} = a_{n - 1} \cdot r$ ,

Where $a_{1}$ is the first term, $n$ is the term you want, and $r$ is the common ratio.

Explicit General Formula

$a_{n} = a_{1} (r)^{n - 1}$

Where $a_{1}$ is the first term, $n$ is the term you want, and $r$ is the common ratio.

Self Check

For the function $J(n)= 180(\frac13)^{n-1}$ , what is the value of the function when $n=5$ ?

$\frac{1}{81}$
$\frac{180}{81}$
20
45

Additional Resources

The $n^{th}$ Term of a Geometric Sequence

The

n^{th}

Term of a Geometric Sequence

General Term (nth Term) of a Geometric Sequence

The general term of a geometric sequence with first term $a_{1}$ and the common ratio $r$ is: $a_{n} = a_{1} r^{n - 1}$ .

Example 1

Write the explicit formula for the sequence given by the terms 3, 2, 43, 89 ...

Solution

Step 1- Write the general formula.

$a_{n} = a_{1} (r^{n - 1})$

Step 2 - Substitute values for the first term, common difference/ratio, term number.

$a_{1} = 3$ , $r = \frac{2}{3}$

$a_{n} = 3 (\frac{2}{3})^{n - 1}$

So, the explicit formula for the sequence 3, 2, 43, 89 ... is $a_{n} = 3 (\frac{2}{3})^{n - 1}$ .

Just like it is with arithmetic sequences, the explicit formula makes finding any term easier to determine - especially term numbers that are very large!

Example 2

Find the $14^{th}$ term of a sequence where the first term is 64 and the common ratio, $r = \frac{1}{2}$ .

Solution

Step 1 - Write the general formula.

$a_{n} = a_{1} (r^{n - 1})$

Step 2 - Substitute values for the first term, common difference/ratio, term number.

$a_{1} = 64$ , $r = \frac{1}{2}$ , $n = 14$

$a_{14} = 64 (\frac{1}{2})^{14 - 1}$

Step 3 - Simplify the expression.

$a_{14} = 64 (\frac{1}{2})^{13}$

$a_{14} = 64 (\frac{1}{8192})$

$a_{14} = \frac{1}{128}$

If the explicit formula for this question was needed, we would not have substituted $n = 14$ and the $n^{th}$ term formula would have been $a_{n} = 64 (\frac{1}{2})^{n - 1}$ .

Try it

The $n^{th}$ Term of a Geometric Sequence

Find the formula for the $n^{th}$ term of a sequence where the first term is 81 and the common ratio is $r = \frac{1}{3}$ .

Compare your answer:

Here is how to find the $11^{th}$ term of the sequence:

Step 1 - Write the general formula.

$a_{n} = a_{1} (r^{n - 1})$

Step 2 - Substitute values for the first term, common difference/ratio, term number.

$a_{11} = 81$ , $r = \frac{1}{3}$

$a_{n} = 81 (\frac{1}{3})^{n - 1}$

4.18.4 Define a Geometric Sequence by the nth Term

Activity

Geometric Sequence Formulas

Self Check

Additional Resources

The nthnth Term of a Geometric Sequence

Try it

The nthnth Term of a Geometric Sequence

The $n^{th}$ Term of a Geometric Sequence

The $n^{th}$ Term of a Geometric Sequence