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Calculus 2 : Types of Series

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Example Question #17 : Arithmetic And Geometric Series

Find the value of the 300th term, or $a_{300}$ , in the following arithmetic series:

$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+...=40+57+74+91+108+...$

Possible Answers:

5115

5123

5125

5120

Correct answer:

5123

Explanation:

3 pieces of information are needed to find the value of a specific term.

First, find the first value, $a_{1}$ .

Second, the value of n, or the total number of terms in the series.

Finally, d, or the common difference, which can be found by calculating $a_{2} -a{1}$ .

With these pieces of information, find the value of the last with the following formula:

$a_{n} = a_{1} + (n-1)*d$

Solution:

$a_{1}=40$

n=300

$d = a_{2}-a_{1}=57-40=17$

$a_{300} = a_{1} + (n-1)*d=40+(300-1)*17=40+5083=5123$

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Example Question #1 : Series Of Constants

Calculate the sum of a geometric series with the following values: n=15 , $a_{1}=4$ , $r=\frac{1}{3}$ . Round the answer to the nearest integer.

Possible Answers:

Correct answer:

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

$S_{n} = \frac{a_{1}*(1-r^n)}{1-r}$ , where n is the number of terms to sum up, r is the common ratio, and $a_{1}$ is the value of the first term.

For this question, we are given all of the information we need.

Solution:

$S_{15}=\frac{a_{1}*(1-r^n)}{1-r}$

$S_{15}=\frac{4*(1-(\frac{1}{3})^{15})}{1-\frac{1}{3}}$

$S_{15}=\frac{4*3(1-\frac{1}{3^{15}})}{2}$

$S_{15}=6*(1-\frac{1}{3^{15}})$

$S_{15}=5.999...$

Rounding, $S_{15}=6$

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Example Question #18 : Arithmetic And Geometric Series

Calculate the sum, rounded to the nearest integer, of the first 20 terms of the following geometric series: 4+6+9+13.5+20.25...

Possible Answers:

26597

26596

26595

26594

Correct answer:

26594

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

$S_{n} = \frac{a_{1}*(1-r^n)}{1-r}$ , where n is the number of terms to sum up, r is the common ratio, and $a_{1}$ is the value of the first term.

We have $a_{1}$ and n and we just need to find r before calculating the sum.

Solution:

$a_{1}=4$

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

$r=\frac{6}{4} = 1.5$

$S_{20}=\frac{a_{1}*(1-r^n)}{1-r}$

$S_{20}=\frac{4*(1-1.5^{20})}{1-1.5}$

$S_{20}=26594$

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Example Question #2 : Series Of Constants

Calculate the sum, rounded to the nearest integer, of the first 16 terms of the following geometric series: 6+12.6+26.46+116.6886+245.04606...

Possible Answers:

780295

780310

780305

780300

Correct answer:

780305

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

$S_{n} = \frac{a_{1}*(1-r^n)}{1-r}$ , where n is the number of terms to sum up, r is the common ratio, and $a_{1}$ is the value of the first term.

We have $a_{1}$ and n and we just need to find r before calculating the sum.

Solution:

$a_{1}=6$

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

$r=\frac{12.6}{6} = 2.1$

$S_{16}=\frac{a_{1}*(1-r^n)}{1-r}$

$S_{16}=\frac{6*(1-2.1^{16})}{1-2.1}$

$S_{16}=780305$

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Example Question #21 : Arithmetic And Geometric Series

Calculate the sum, rounded to the nearest integer, of the first 9 terms of the following geometric series: 2+9.2+42.32+895.4921+4119.25952...

Possible Answers:

512328

512329

512327

512330

Correct answer:

512327

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

$S_{n} = \frac{a_{1}*(1-r^n)}{1-r}$ , where n is the number of terms to sum up, r is the common ratio, and $a_{1}$ is the value of the first term.

We have $a_{1}$ and n and we just need to find r before calculating the sum.

Solution:

$a_{1}=2$

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

$r=\frac{9.2}{2} = 4.6$

$S_{9}=\frac{a_{1}*(1-r^n)}{1-r}$

$S_{9}=\frac{2*(1-4.6^{9})}{1-4.6}$

$S_{9}=512327$

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Example Question #22 : Arithmetic And Geometric Series

Calculate the sum, rounded to the nearest integer, of the first 100 terms of the following geometric series: 16+13.76+11.8336+10.176896+8.75213056...

Possible Answers:

115

114

116

117

Correct answer:

114

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

$S_{n} = \frac{a_{1}*(1-r^n)}{1-r}$ , where n is the number of terms to sum up, r is the common ratio, and $a_{1}$ is the value of the first term.

We have $a_{1}$ and n and we just need to find r before calculating the sum.

Solution:

$a_{1}=16$

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

$r=\frac{13.76}{16} = .86$

$S_{100}=\frac{a_{1}*(1-r^n)}{1-r}$

$S_{100}=\frac{16*(1-.86^{100})}{1-.86}$

$S_{100}=144$

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Example Question #23 : Arithmetic And Geometric Series

Calculate the sum of the following infinite geometric series:

$\sum_{k=0}^{\infty}19\left(\frac{6}{15}\right)^k$

Possible Answers:

$\frac{95}{3}$

$\frac{284}{9}$

$\frac{283}{9}$

$\frac{286}{9}$

Correct answer:

$\frac{95}{3}$

Explanation:

This is an infinite geometric series.

The sum of an infinite geometric series can be calculated with the following formula,

$S = \frac{a_{1}}{1-r}$ , where $a_{1}$ is the first value of the summation, and r is the common ratio.

Solution:

Value of $a_{1}$ can be found by setting k = 0

$a_{1}=19$

r is the value contained in the exponent

$r=\frac{6}{15}$

$S=\frac{a_{1}}{1-r}$

$S=\frac{19}{1-\frac{6}{15}} = \frac{19}{\frac{9}{15}}$

$S=\frac{285}{9}=\frac{95}{3}$

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Example Question #201 : Series In Calculus

Calculate the sum of a geometric series with the following values:

n=10 , $a_{1}=12$ , $r=\frac{1}{4}$ ,

rounded to the nearest integer.

Possible Answers:

Correct answer:

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

$S_{n} = \frac{a_{1}*(1-r^n)}{1-r}$ , where n is the number of terms to sum up, r is the common ratio, and $a_{1}$ is the value of the first term.

For this question, we are given all of the information we need.

Solution:

$S_{10}=\frac{a_{1}*(1-r^n)}{1-r}$

$S_{10}=\frac{12*(1-(\frac{1}{4})^{10})}{1-\frac{1}{4}}$

$S_{10}=\frac{12*(1-\frac{1}{4^{10}}}{\frac{3}{4}} = \frac{48}{3}(1-\frac{1}{4^{10}})$

$S_{10}=16*(1-\frac{1}{4^{10}})$

$S_{10}=15.999$

Rounding, $S_{10}=16$

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Example Question #21 : Arithmetic And Geometric Series

Calculate the sum of a geometric series with the following values:

n=40 , $a_{1}=5,$ $r=\frac{2}{7}$ ,

rounded to the nearest integer.

Possible Answers:

Correct answer:

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

$S_{n} = \frac{a_{1}*(1-r^n)}{1-r}$ , where n is the number of terms to sum up, r is the common ratio, and $a_{1}$ is the value of the first term.

For this question, we are given all of the information we need.

Solution:

$S_{40}=\frac{a_{1}*(1-r^n)}{1-r}$

$S_{40}=\frac{5*(1-(\frac{2}{7})^{40})}{1-\frac{2}{7}}$

$S_{40}=\frac{5*(1-(\frac{2}{7})^{40})}{\frac{5}{7}} = \frac{5*7(1-(\frac{2}{7})^{40})}{5}$

$S_{40}=7*(1-(\frac{2}{7})^{40})$

$S_{40}=6.99999...$

Rounding, $S_{40}=7$

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Example Question #4 : Geometric Series

Calculate the sum of a geometric series with the following values:

n=10 , $a_{1}=11$ , $r=\frac{2}{20}$ ,

rounded to the nearest integer.

Possible Answers:

Correct answer:

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

$S_{n} = \frac{a_{1}*(1-r^n)}{1-r}$ , where n is the number of terms to sum up, r is the common ratio, and $a_{1}$ is the value of the first term.

For this question, we are given all of the information we need.

Solution:

$S_{10}=\frac{a_{1}*(1-r^n)}{1-r}$

$S_{10}=\frac{11*(1-(\frac{2}{20})^{10})}{1-\frac{2}{20}}$

$S_{10}=\frac{11*(1-(\frac{1}{10})^{10}}{\frac{9}{10}} = \frac{11*10*(1-\frac{1}{10^{10}})}{9}$

$S_{10}=\frac{110}{9}(1-\frac{1}{10^{10}})$

$S_{10}=12.2222...$

Rounding, $S_{10}=12$

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Calculus 2 : Types of Series

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #17 : Arithmetic And Geometric Series

Example Question #1 : Series Of Constants

Example Question #18 : Arithmetic And Geometric Series

Example Question #2 : Series Of Constants

Example Question #21 : Arithmetic And Geometric Series

Example Question #22 : Arithmetic And Geometric Series

Example Question #23 : Arithmetic And Geometric Series

Example Question #201 : Series In Calculus

Example Question #21 : Arithmetic And Geometric Series

Example Question #4 : Geometric Series

All Calculus 2 Resources

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