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Algebra 1

4.18.4 Define a Geometric Sequence by the nth Term

Algebra 14.18.4 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 W n = 284 ( 1.04 ) n , where W W is the projected wildcat population n n years after 2015. Explain where the different factors in her expression came from.

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 n = − 2 on your paper and check your answers.

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

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

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

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

Remember that a recursive formula defines terms using one or more of the previous terms. If you need to calculate the 100 t h 100 t h term or the 500 t h 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 n = 100 or n = 500 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 · r a n = a n − 1 · r ,

Where a 1 a 1 is the first term, n n is the term you want, and r r is the common ratio.

Explicit General Formula

a n = a 1 ( r ) n − 1 a n = a 1 ( r ) n − 1

Where a 1 a 1 is the first term, n n is the term you want, and r r is the common ratio.

Self Check

For the function J ( n ) = 180 ( 1 3 ) n − 1 , what is the value of the function when n = 5 ?

  1. 1 81
  2. 180 81
  3. 20
  4. 45

Additional Resources

The nthnth Term of a Geometric Sequence

The nthnth 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 a 1 and the common ratio r r is: a n = a 1 r n − 1 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 ) a n = a 1 ( r n − 1 )

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

a 1 = 3 a 1 = 3 , r = 2 3 r = 2 3

a n = 3 ( 2 3 ) n − 1 a n = 3 ( 2 3 ) n − 1

So, the explicit formula for the sequence 3, 2, 43, 89 ... is a n = 3 ( 2 3 ) n − 1 a n = 3 ( 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 14th14th term of a sequence where the first term is 64 and the common ratio, r = 1 2 r = 1 2 .

Solution 

Step 1 - Write the general formula.

a n = a 1 ( r n − 1 ) a n = a 1 ( r n − 1 )

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

a 1 = 64 a 1 = 64 , r = 1 2 r = 1 2 , n = 14 n = 14

a 14 = 64 ( 1 2 ) 14 − 1 a 14 = 64 ( 1 2 ) 14 − 1

Step 3 - Simplify the expression.

a 14 = 64 ( 1 2 ) 13 a 14 = 64 ( 1 2 ) 13

a 14 = 64 ( 1 2 ) 13 a 14 = 64 ( 1 2 ) 13

a 14 = 64 ( 1 8192 ) a 14 = 64 ( 1 8192 )

a 14 = 1 128 a 14 = 1 128

If the explicit formula for this question was needed, we would not have substituted n = 14 n = 14 and the nthnth term formula would have been a n = 64 ( 1 2 ) n − 1 a n = 64 ( 1 2 ) n − 1 .

Try it

The nthnth Term of a Geometric Sequence

Find the formula for the nthnth term of a sequence where the first term is 81 and the common ratio is r = 1 3 r = 1 3 .

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