12.3 Geometric Sequences and Series

Before you get started, take this readiness quiz.

Simplify: $\frac{24}{32} .$
If you missed this problem, review the reference.

Evaluate: ⓐ $3^{4}$ ⓑ $\left(\frac{1}{2}\right)^{4} .$
If you missed this problem, review the reference.

If $f \left(x\right) = 4 \cdot 3^{x} ,$ find ⓐ $f \left(1\right)$ ⓑ $f \left(2\right)$ ⓒ $f \left(3\right) .$
If you missed this problem, review the reference.

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

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

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

ⓒ $27 , 9 , 3 , 1 , \frac{1}{3} , \frac{1}{9} , \ldots$

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

ⓐ $7 , 21 , 63 , 189 , 567 , 1,701 , \ldots$

ⓑ $64 , 16 , 4 , 1 , \frac{1}{4} , \frac{1}{16} , \ldots$

ⓒ $2 , 4 , 12 , 48 , 240 , 1,440 , \ldots$

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

ⓐ $−150 , −30 , −15 , −5 , - \frac{5}{2} , 0 , \ldots$

ⓑ $5 , 10 , 20 , 40 , 80 , 160 , \ldots$

ⓒ $8 , 4 , 2 , 1 , \frac{1}{2} , \frac{1}{4} , \ldots$

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

Write the first five terms of the sequence where the first term is 7 and the common ratio is $r = −3 .$

Write the first five terms of the sequence where the first term is 6 and the common ratio is $r = −4 .$

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

Find the thirteenth term of a sequence where the first term is 81 and the common ratio is $r = \frac{1}{3} .$

Find the twelfth term of a sequence where the first term is 256 and the common ratio is $r = \frac{1}{4} .$

Find the twelfth term of the sequence 3, 6, 12, 24, 48, 96, … Find the general term for the sequence.

Find the ninth term of the sequence 6, 18, 54, 162, 486, 1,458, … Then find the general term for the sequence.

Find the eleventh term of the sequence 7, 14, 28, 56, 112, 224, … Then find the general term for the sequence.

Find the sum of the first 20 terms of the geometric sequence 7, 14, 28, 56, 112, 224, …

Find the sum of the first 20 terms of the geometric sequence 3, 6, 12, 24, 48, 96, …

Find the sum of the first 20 terms of the geometric sequence 6, 18, 54, 162, 486, 1,458, …

Find the sum: $\sum_{i = 1}^{15} 6 \left(2\right)^{i} .$

Find the sum: $\sum_{i = 1}^{10} 5 \left(2\right)^{i} .$

Find the sum of the infinite geometric series $54 + 18 + 6 + 2 + \frac{2}{3} + \frac{2}{9} + \ldots$

Find the sum of the infinite geometric series $48 + 24 + 12 + 6 + 3 + \frac{3}{2} + \ldots$

Find the sum of the infinite geometric series $64 + 16 + 4 + 1 + \frac{1}{4} + \frac{1}{16} + \ldots$

Write the repeating decimal $0. \overset{–}{5}$ as a fraction.

Write the repeating decimal $0. \overset{–}{4}$ as a fraction.

Write the repeating decimal $0. \overset{–}{8}$ as a fraction.

The government has decided to give a $1,000 tax rebate to each household in order to stimulate the economy. The government statistics say that each household will spend 80% of the rebate in goods and services. The businesses and individuals who benefitted from that 80% will then spend 80% of what they received and so on. The result is called the multiplier effect. What is the total effect of the r…

What is the total effect on the economy of a government tax rebate of $1,000 to each household in order to stimulate the economy if each household will spend 90% of the rebate in goods and services?

What is the total effect on the economy of a government tax rebate of $500 to each household in order to stimulate the economy if each household will spend 85% of the rebate in goods and services?

New parents decide to invest $100 per month in an annuity for their baby daughter. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at her eighteenth birthday?

New grandparents decide to invest $200 per month in an annuity for their grandson. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at his twenty-first birthday?

Arturo just got his first full-time job after graduating from college at age 27. He decided to invest $200 per month in an IRA (an annuity). The interest on the annuity is 8%, which is compounded monthly. How much will be in the Arturo’s account when he retires at his sixty-seventh birthday?

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

$3 , 12 , 48 , 192 , 768 , 3072 , \ldots$

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

$72 , 36 , 18 , 9 , \frac{9}{2} , \frac{9}{4} , \ldots$

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

$−3 , 6 , −12 , 24 , −48 , 96 , \ldots$

Determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

$48 , 24 , 12 , 6 , 3 , \frac{3}{2} , \ldots$

Determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

$−7 , −2 , 3 , 8 , 13 , 18 , \ldots$

Determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

$\frac{1}{2} , \frac{1}{4} , \frac{1}{8} , \frac{1}{16} , \frac{1}{32} , \frac{1}{64} , \ldots$

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

$a_{1} = 4$ and $r = 3$

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

$a_{1} = −4$ and $r = −2$

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

$a_{1} = 27$ and $r = \frac{1}{3}$

Find the indicated term of a sequence where the first term and the common ratio is given.

Find $a_{11}$ given $a_{1} = 8$ and $r = 3 .$

Find the indicated term of a sequence where the first term and the common ratio is given.

Find $a_{10}$ given $a_{1} = −6$ and $r = −2 .$

Find the indicated term of a sequence where the first term and the common ratio is given.

Find $a_{10}$ given $a_{1} = 100,000$ and $r = 0.1 .$

Find the indicated term of the given sequence. Find the general term for the sequence.

Find $a_{9}$ of the sequence, $9 , 18 , 36 , 72 , 144 , 288 , \ldots$

Find the indicated term of the given sequence. Find the general term for the sequence.

Find $a_{15}$ of the sequence, $−486 , 162 , −54 , 18 , −6 , 2 , \ldots$

Find the indicated term of the given sequence. Find the general term for the sequence.

Find $a_{10}$ of the sequence, $1 , 0.1 , 0.01 , 0.001 , 0.0001 , 0.00001 , \ldots$

Find the sum of the first fifteen terms of each geometric sequence.

$8 , 24 , 72 , 216 , 648 , 1944 , \ldots$

Find the sum of the first fifteen terms of each geometric sequence.

$−6 , 12 , −24 , 48 , −96 , 192 , \ldots$

Find the sum of the first fifteen terms of each geometric sequence.

$81 , 27 , 9 , 3 , 1 , \frac{1}{3} , \ldots$

Find the sum of the geometric sequence.

$\sum_{i = 1}^{15} \left(2\right)^{i}$

Find the sum of the geometric sequence.

$\sum_{i = 1}^{9} 4 \left(2\right)^{i}$

Find the sum of the geometric sequence.

$\sum_{i = 1}^{10} 9 \left(\frac{1}{3}\right)^{i}$

Find the sum of each infinite geometric series.

$1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} + \frac{1}{729} + \ldots$

Find the sum of each infinite geometric series.

$6 - 2 + \frac{2}{3} - \frac{2}{9} + \frac{2}{27} - \frac{2}{81} + \ldots$

Find the sum of each infinite geometric series.

$6 + 12 + 24 + 48 + 96 + 192 + \ldots$

Find the sum of each infinite geometric series.

$1,024 + 512 + 256 + 128 + 64 + 32 + \ldots$

Write each repeating decimal as a fraction.

$0. \overset{–}{3}$

Write each repeating decimal as a fraction.

$0. \overset{–}{7}$

Write each repeating decimal as a fraction.

$0. \overset{—}{45}$

Solve the problem.

Find the total effect on the economy of each government tax rebate to each household in order to stimulate the economy if each household will spend the indicated percent of the rebate in goods and services.

Solve the problem.

Berenice just got her first full-time job after graduating from college at age 30. She decided to invest $\$ 500$ per quarter in an IRA (an annuity). The interest on the annuity is $7 \%$ which is compounded quarterly (4 times a year). How much will be in the Berenice’s account when she retires at age 65?

Solve the problem.

Myra just got her first full-time job after graduating from college. She plans to get a master’s degree, and so is depositing $\$ 2,500$ a year from her year-end bonus into an annuity. The annuity pays $6.5 \%$ per year and is compounded yearly. How much will she have saved in five years to pursue her master’s degree?

In your own words, explain how to find the general term of a geometric sequence.

In your own words, explain how to determine if an infinite geometric series has a sum and how to find it.