The formula for the

nth term involves four variables:a,_{n}a_{1},r, andn. If we know the value of any three of them, we can find the value of the fourth.

Explicit geometric sequencesalso have a formula for finding any term in a sequence.

a=_{n}a_{1}r^{(n-1)}

a= the term in the sequence you are trying to find (_{n}nrepresents the desired term number)

a_{1}= the first term in the sequence

r= the common ratio

Example 1. What is the 15th term of the geometric sequence -9, 27, -81, … ?

Solution:

Calculate the common ratio.

The common ratio is -3.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{15}=-9⋅(-3)

^{(15-1)}

=-9⋅(-3)

^{14}

=-43,046,721

The 15th term of the sequence is -43,046,721.

Ex2. What is the 10th term of the geometric sequence 6, -24, 96, … ?

Solution:

Calculate the common ratio.

The common ratio is -4.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{10}=6⋅(-4)

^{(10-1)}

=6⋅(-4)

^{9}

=-1,572,864

The 10th term of the sequence is -1,572,864.

Ex3. Find the eleventh term of the sequence 3, -6, 12, -24, …

(A). 1024 (B). 3072 (C). 33 (D). -6144

Solution:

Calculate the common ratio.

The common ratio is -2.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{11}=3(-2)

^{(11-1)}

=3(-2)

^{10}

*a*

_{11}=3072

The eleventh term of the sequence is 3072. Choice B is the correct answer.

Ex4 — Ex10: **Find the specified term for each geometric sequence or sequence with the given characteristics**

Ex4. *a*_{9} for 60, 30, 15, … .

Solution:

First, find the common ratio.

15÷30=½

Use the formula for the

*n*th term of a geometric sequence to find

*a*

_{9}.

Ex5. *a*_{4} for 7, 14, 28, … .

Solution:

First, find the common ratio.

28÷14=2

Use the formula for the

*n*th term of a geometric sequence to find

*a*

_{4}.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{4}=7⋅2

^{(4-1)}

*a*

_{4}=7⋅8

*a*

_{4}=56

Ex6. *a*_{5} for 3, 1, ⅓, … .

Solution:

First, find the common ratio.

⅓÷1=⅓

Use the formula for the

*n*th term of a geometric sequence to find

*a*

_{5}.

Ex7. *a*_{6} for 540, 90, 15, … .

Solution:

First, find the common ratio.

15÷90=⅙

Use the formula for the

*n*th term of a geometric sequence to find

*a*

_{6}.

Ex8. *a*_{7} if *a*_{3}=24 and *r*=0.5.

Solution:

*a*

_{7}=

*a*

_{3}

*r*

^{(7-3)}

*a*

_{7}=24⋅½

^{4}

*a*

_{7}=1.5

Ex9. *a*_{6} if *a*_{3}=32 and *r*=-0.5.

Solution:

*a*

_{6}=

*a*

_{3}

*r*

^{(6-3)}

*a*

_{6}=32⋅(-½)

^{3}

*a*

_{6}=32⋅(-⅛)

*a*

_{6}=-4

Ex10. *a*_{8} if *a*_{1}=4096 and *r*=¼.

Solution:

Use the formula for the *n*th term of a geometric sequence to find *a*_{8}.

Ex11 — Ex14: **Find the indicated term of the geometric sequence**

Ex11. The 10th term; 6, 18, 54, 162, …

Solution:

First develop a formula for *a _{n}*; first term is 6 and common ratio is 3.

*a*=

_{n}*x*(3

^{n})

*a*

_{1}=

*x*(3

^{1})

6=3

*x*

2=

*x*

Now, find the 10th term:

*a*

_{10}=2(3

^{10})=118098

Answer: 118098

Ex12. The 9th term; 5, 20, 80, 160, …

Solution:

First develop a formula for *a _{n}*; first term is 5 and common ratio is 4

formula for

*a*=5/4 (4

_{n}^{n}).

Now, find the 9th term:

*a*

_{9}=5/4 (4

^{9})=327680

Answer: 327680

Ex13. The 14th term; 200, 100, 50, 25, …

Solution:

First develop a formula for *a _{n}*; first term 200, common ratio ½.

*a*=

_{n}*x*(½

^{n})

*a*

_{1}=

*x*(½

^{1})

200=½

*x*

2⋅200=2⋅½

*x*

*x*=400

formula for

*a*=400(½

_{n}^{n}).

Now, find the 14th term:

*a*

_{14}=400(½

^{14})=25/1024

Answer: 25/1024

Ex14. The 20th term; 39366, 13122, 4374, …

Solution:

First develop a formula for *a _{n}*; first term 39366 common ratio ⅓.

*a*=

_{n}*x*(⅓

^{n})

*a*

_{1}=

*x*(⅓

^{1})

39366=⅓

*x*

3⋅39366=3⋅⅓

*x*

*x*=118098

*a*=118098(⅓

_{n}^{n})

now I can find

*a*

_{20}=118098(⅓

^{20})=2/59049

Answer: 2/59049

Ex15. Find 4th term in the geometric sequence 5, 10, 20, …

Solution:

*a*=5,

*r*=10/5=2,

*a*=?

_{n}*a*=

_{n}*a*

*r*

^{(n-1)}

*a*

_{4}=5⋅2

^{(4-1)}=5×8=40

Ex16. Determine the 10th term of the sequence: 3, 6, 12, …

Solution:

*a*=3,

*r*=6/3=12/6=2

*a*=

_{n}*a*

*r*

^{(n-1)}

*a*

_{10}=3⋅2

^{(10-1)}=3⋅2

^{9}=3⋅512=1536

Ex17. Find the 7th term in the following sequence: 6, 18, 54, 162, …

Solution:

Finding the common ratio can be harder than finding the common difference. One way to find it is the divide each term by the term before it.

18÷6=3, 54÷18=3, 162÷54=3. So the common ratio is 3.

*a*

_{7}=6⋅3

^{((7-1)))}

*a*

_{7}=6⋅3

^{6}=6×729

*a*

_{7}=4,374

So the 7th term of the sequence is 4,374.

Ex18. Find the 8th term in the following sequence: 96, 48, 24, 12, 6, …

Solution:

To find the common ratio, divide each term by the one before it.

48÷96=½, 24÷48=½, 12÷24=½. The common ratio is ½.

The 8th term of the sequence is 0.75.

Ex19. What is the 10th term of the sequence 4, 12, 36, 108, 324, … ?

Solution:

The sequence 4, 12, 36, 108, 324, … is a geometric sequence with *a*_{1}=4 and *r*=3.Therefore,

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{10}=4⋅3

^{9}=78732

Use a calculator for the computation.

Ex20: **Find the Next Term**

Test-Taking Tip

Since the terms of this sequence are increasing, the missing term must be greater than 125. You can immediately eliminate 75 as a possible answer.

**Multiple-Choice Test Item**

Find the missing term in the geometric sequence: 8, 20, 50, 125, _

(A). 75 (B). 200 (C). 250 (D). 312.5

Solution:

Since 20/8=2.5, 50/20=2.5, and 125/50=2.5, the sequence has a common ratio of 2.5.

**Solve the Test Item**

To find the missing term, multiply the last given term by 2.5:

The answer is D.

Ex21. The first term of a geometric series is 1 and the common ratio is 9. What is the 8th term of the sequence?

Solution:

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{8}=1⋅9

^{(8-1)}

=1⋅9

^{7}

=4,782,969

The 8th term of the sequence is 4,782,969.

Ex22. The first term of a geometric series is 2 and the common ratio is 4. What is the 14th term of the sequence?

Solution:

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{14}=2⋅4

^{(14-1)}

=2⋅4

^{13}

=134, 217, 728

The 14th term of the sequence is 134, 217, 728.

Ex23: F**inding a missing term**

Find the first term of a geometric sequence whose fourth term is 8 and whose common ratio is ½.

Solution:

Let *a*_{4}=8, *r*=½, andn *n*=4 in the formula *a _{n}*=

*a*

_{1}

*r*

^{(n-1)}.

*a*

_{1}⋅½

^{(4-1)}

8=

*a*

_{1}⋅⅛

64=

*a*

_{1}

So the first term is 64.

Ex24. If *r*=4 and *a*_{8}=100, what is the first term of the geometric sequence?

Solution:

Substitute *a*_{8}=100, *r*=4, and *n*=8 into the formula for the *n*th term of a geometric sequence to find the *a*_{1}.

Ex25: **Find a Term Given the Fourth Term and the Ratio**

Find the tenth term of a geometric sequence for which *a*_{4}=108 and *r*=3.

Solution:

First, find the value of *a*_{1}. Formula for *n*th term, *a _{n}*=

*a*

_{1}⋅

*r*

^{(n-1)}.

*n*=4,

*r*=3

*a*

_{4}=

*a*

_{1}⋅3

^{(4-1)}

*a*

_{4}=108

108=27

*a*

_{1}

Divide each side by 27. Formula for

*n*th term,

*a*=

_{n}*a*

_{1}⋅

*r*

^{(n-1)}.

*n*=10,

*a*

_{1}=4,

*r*=3

*a*=

_{n}*a*

_{1}⋅

*r*

^{(n-1)}

*a*

_{10}=4⋅3

^{(10-1)}

*a*

_{10}=78, 732

Ex26. Determine the 12th term of a geometric sequence whose 8th term is 192 and the common ratio is 2.

Solution: We have

Ex27. Find the eighth term of a geometric sequence for which *a*_{3}=81 and *r*=3.

Solution:

Because *a*_{3}=81, the third term in the sequence is 81. To find the eighth term of the sequence, you need to find the 1st term of the sequence. Use the *n*th term of a Geometric Sequence formula.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{3}=

*a*

_{1}⋅3

^{(3-1)}

81=

*a*

_{1}⋅9

*a*

_{1}=9

Then the first term

*a*

_{1}is 9.

Use

*a*

_{1}to find the eighth term of the sequence.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{8}=

*a*

_{1}⋅3

^{(8-1)}

=9⋅3

^{7}

=19,683

The eighth term of the geometric sequence is 19,683.

Ex28. Find the 12th term of a geometric sequence whose 8th term is 192 and the common ratio is 2.

Solution:

Common ratio, *r*=2

Let *a* be the first term of the geometric sequence.

Ex29: **Find a Particular Term**

Find the eighth term of a geometric sequence for which *a*_{1}=-3 and *r*=-2.

Solution:

Formula for *n*th term, *a _{n}*=

*a*

_{1}⋅

*r*

^{(n-1)}.

*n*=8,

*a*

_{1}=-3,

*r*=-2

*a*

_{8}=(-3)⋅(-2)

^{(8-1)}

*a*

_{8}=(-3)⋅(-128)

*a*

_{8}=384

The eighth term is 384.

Ex30. Find the sixth term of a geometric sequence with a first term of 9 and a common ratio of 2.

Solution:

Use the formula for the *n*th term of a geometric sequence to find the *a*_{6}.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{6}=9⋅2

^{(6-1)}

*a*

_{6}=288

Ex31: **Using the Formula for the General Term of a Geometric sequence**

Find the eighth term of the geometric sequence whose first term is -4 and whose common ratio is -2.

Solution:

To find the eighth term, *a*_{8}, we replace *n* in the formula with 8, *a*_{1} with -4, and *r* with

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{8}=-4(-2)

^{(8-1)}=-4(-2)

^{7}=-4(-128)=512

The eighth term is 512. We can check this result by writing the first eight terms of the Growth sequence:

Ex32. (Biology) A certain bacteria grows at a rate of 3 cells every 2 minutes. If there were 260 cells initially, how many are there after 21 minutes?

Solution:

Given, *a*_{1}=260, *n*=21 and *r*=3/2

Answer: 864,567

Ex33. (Biology) A virus goes through a computer, infecting the files. If one file was infected initially and the total number of files infected doubles every minute, how many files will be infected in 20 minutes?

Solution:

Given, *a*_{1}=1, *r*=200% or 2, *n*=20.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{20}=1⋅2

^{(20-1)}

=524,288

Answer: 524,288

Ex34. (Short Response) Elisa has a savings account. She withdraws half of the contents every year. After 4 years, she has $2000 left. How much did she have in the savings account originally?

Solution:

Given *n*=4, *a*_{4}=2000, *r*=0.5

She have invested (

*a*

^{0}) $32, 000.

Answer: $32, 000

Ex35. (Sense-Making)

A certain drug has a half-life of 8 hours after it is administered to a patient. What percent of the drug is still in the patient’s system after 24 hours?

Solution:

Given *a*_{1}=100% or 1 and *r*=½ and *n*=4.

*a*=

_{n}*a*

_{1}

*r*

^{(n-1)}

*a*

_{4}=1⋅½

^{3}

=⅛

=0.125 or 12.5%

After 24 hours, 12.5% of the drug is still in the patient’s system.

Answer: 12.5%