finite geometric series assignment quizlet

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24.1: Finite Geometric Series

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• Page ID 54480

• Thomas Tradler and Holly Carley
• CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

We now study another sequence, the geometric sequence, which will be analogous to our study of the arithmetic sequence in section 23.2 . We have already encountered examples of geometric sequences in Example 23.1.1 (b) . A geometric sequence is a sequence for which we multiply a constant number to get from one term to the next, for example:

\[5,\underset{\times 4}{\hookrightarrow } 20,\underset{\times 4}{\hookrightarrow } 80,\underset{\times 4}{\hookrightarrow } 320,\underset{\times 4}{\hookrightarrow } 1280, \dots \nonumber \]

Definition: Geometric Sequence

A sequence \(\{a_n\}\) is called a geometric sequence , if any two consecutive terms have a common ratio \(r\) . The geometric sequence is determined by \(r\) and the first value \(a_1\) . This can be written recursively as:

\[a_n=a_{n-1}\cdot r \quad \quad \text{for }n\geq 2 \nonumber \]

Alternatively, we have the general formula for the \(n\) th term of the geometric sequence:

\[\label{EQU:geometric-sequence-general-term} \boxed{a_n=a_1\cdot r^{n-1}}\]

Example \(\PageIndex{1}\)

Determine if the sequence is a geometric, or arithmetic sequence, or neither or both. If it is a geometric or arithmetic sequence, then find the general formula for \(a_n\) in the form \(\ref{EQU:geometric-sequence-general-term}\) or [EQU:arithmetic-sequence-general-term] .

• \(3, 6, 12, 24, 48, \dots\)
• \(100, 50, 25, 12.5, \dots\)
• \(700, -70, 7, -0.7, 0.07, \dots\)
• \(2, 4, 16, 256, \dots\)
• \(3, 10, 17, 24, \dots\)
• \(-3, -3, -3, -3, -3, \dots\)
• \(a_n=\left(\dfrac{3}{7}\right)^n\)
• \(a_n=n^2\)
• Calculating the quotient of two consecutive terms always gives the same number \(6\div 3=2\) , \(12\div 6=2\) , \(24\div 12=2\) , etc. Therefore the common ratio is \(r=2\) , which shows that this is a geometric sequence. Furthermore, the first term is \(a_1=3\) , so that the general formula for the \(n\) th term is \(a_n=3\cdot 2^{n-1}\) .
• We see that the common ratio between two terms is \(r=\dfrac 1 2\) , so that this is a geometric sequence. Since the first term is \(a_1=100\) , we have the general term \(a_n=100\cdot \left(\dfrac 1 2\right)^{n-1}\) .
• Two consecutive terms have a ratio of \(r=-\dfrac 1 {10}\) , and the first term is \(a_1=700\) . The general term of this geometric sequence is \(a_n=700\cdot \left(-\dfrac 1 {10}\right)^{n-1}\) .
• The quotient of the first two terms is \(4\div 2=2\) , whereas the quotient of the next two terms is \(16\div 4=4\) . Since these quotients are not equal, this is not a geometric sequence. Furthermore, the difference between the first two terms is \(4-2=2\) , and the next two terms have a difference \(16-4=12\) . Therefore, this is also not an arithmetic sequence.
• The quotient of the first couple of terms is not equal \(\dfrac{10}{3}\neq \dfrac{17}{10}\) , so that this is not a geometric sequence. The difference of any two terms is \(7=10-3=17-10=24-17\) , so that this is part of an arithmetic sequence with common difference \(d=7\) . The general formula is \(a_n=a_1+d\cdot (n-1)=3+7\cdot (n-1)\) .
• The common ratio is \(r=(-3)\div (-3)=1\) , so that this is a geometric sequence with \(a_n=(-3)\cdot 1^{n-1}\) . On the other hand, the common difference is \((-3)-(-3)=0\) , so that this is also an arithmetic sequence with \(a_n=(-3)+0\cdot (n-1)\) . Of course, both formulas reduce to the simpler expression \(a_n=-3\) .
• Writing the first couple of terms in the sequence \(\left\{\left(\dfrac 3 7 \right)^n\right\}\) , we obtain:

\[\left(\dfrac 3 7 \right)^1, \left(\dfrac 3 7 \right)^2, \left(\dfrac 3 7 \right)^3, \left(\dfrac 3 7 \right)^4, \left(\dfrac 3 7 \right)^5, \dots \nonumber \]

Thus, we get from one term to the next by multiplying \(r=\dfrac 3 7\) , so that this is a geometric sequence. The first term is \(a_1=\dfrac 3 7\) , so that \(a_n=\dfrac 3 7 \cdot \left(\dfrac 3 7 \right)^{n-1}\) . This is clearly the given sequence, since we may simplify this as

\[a_n=\dfrac 3 7 \cdot \left(\dfrac 3 7 \right)^{n-1} =\left(\dfrac 3 7 \right)^{1+n-1}=\left(\dfrac 3 7 \right)^{n} \nonumber \]

• We write the first terms in the sequence \(\{n^2\}_{n\geq 1}\) :

\[1, 4, 9, 16, 25, 36, 49, \dots \nonumber \]

Calculating the quotients of consecutive terms, we get \(4\div 1 =4\) and \(9\div 4=2.25\) , so that this is not a geometric sequence. Also the difference of consecutive terms is \(4-1=3\) and \(9-4=5\) , so that this is also not an arithmetic sequence.

Example \(\PageIndex{2}\)

Find the general formula of a geometric sequence with the given property.

• \(r=4\) , and \(a_5=6400\)
• \(a_1=\dfrac{2}{5}\) , and \(a_{4}=-\dfrac{27}{20}\)
• \(a_{5}=216\) , \(a_{7}=24\) , and \(r\) is positive
• Since \(\{a_n\}\) is a geometric sequence, it is \(a_n=a_1\cdot r^{n-1}\) . We know that \(r=4\) , so we still need to find \(a_1\) . Using \(a_5=64000\) , we obtain:

\[6400=a_5=a_1\cdot 4^{5-1}=a_1\cdot 4^4=256\cdot a_1 \,\, \stackrel{(\div 256)}{\implies} \,\, a_1=\dfrac{6400}{256}=25 \nonumber \]

The sequence is therefore given by the formula, \(a_n=25\cdot 4^{n-1}\) .

• The geometric sequence \(a_n=a_1\cdot r^{n-1}\) has \(a_1=\dfrac{2}{5}\) . We calculate \(r\) using the second condition.

\[\begin{aligned} -\dfrac{27}{20}=a_4=a_1\cdot r^{4-1}=\dfrac{2}{5}\cdot r^3 & \stackrel{(\times \frac{5}{2})}{\implies} & r^3= -\dfrac{27}{20}\cdot \dfrac 5 2= -\dfrac{27}{4}\cdot \dfrac 1 2=\dfrac{-27}8 \\ & \stackrel{(\text{take }\sqrt[3]{\,\,})}{\implies} & r=\sqrt[3]{\dfrac{-27}8}=\dfrac{\sqrt[3]{-27}}{\sqrt[3]{8}}=\dfrac{-3}{2}\end{aligned}\nonumber \]

Therefore, \(a_n=\dfrac 2 5 \cdot \left(\dfrac {-3} 2\right)^{n-1}\) .

• The question does neither provide \(a_1\) nor \(r\) in the formula \(a_n=a_1\cdot r^{n-1}\) . However, we obtain two equations in the two variables \(a_1\) and \(r\) :

\[\left\{\begin{array} { c } { 2 1 6 = a _ { 5 } = a _ { 1 } \cdot r ^ { 5 - 1 } } \\ { 2 4 = a _ { 7 } = a _ { 1 } \cdot r ^ { 7 - 1 } } \end{array} \quad \Longrightarrow \left\{\begin{array}{c} 216=a_{1} \cdot r^{4} \\ 24=a_{1} \cdot r^{6} \end{array}\right.\right. \nonumber \]

In order to solve this, we need to eliminate one of the variables. Looking at the equations on the right, we see dividing the top equation by the bottom equation cancels \(a_1\) .

\[\dfrac{216}{24}=\dfrac{a_1\cdot r^4}{a_1\cdot r^6} \,\, \implies \,\, \dfrac{9}{1}=\dfrac{1}{r^2} \,\,\stackrel{(\text{take reciprocal})}\implies \,\, \dfrac{1}{9}=\dfrac{r^2}{1} \,\, \implies \,\, r^2= \dfrac 1 9 \nonumber \]

To obtain \(r\) we have to solve this quadratic equation. In general, there are in fact two solutions:

\[r=\pm\sqrt{\dfrac 1 9}=\pm\dfrac 1 3 \nonumber \]

Since the problem states that \(r\) is positive, we see that we need to take the positive solution \(r=\dfrac 1 3\) . Plugging \(r=\dfrac 1 3\) back into either of the two equations, we may solve for \(a_1\) . For example, using the first equation \(a_5=216\) , we obtain:

\[\begin{aligned} && 216=a_5=a_1\cdot \bigg(\dfrac 1 3\bigg)^{5-1}=a_1\cdot \bigg(\dfrac 1 3\bigg)^{4}=a_1\cdot \dfrac 1 {3^4}=a_1\cdot \dfrac 1 {81} \\ && \quad \stackrel{(\times 81)}\implies \quad a_1=81\cdot 216 = 17496\end{aligned} \nonumber \]

So, we finally arrive at the general formula for the \(n\) th term of the geometric sequence, \(a_n=17496\cdot \left(\dfrac 1 3\right)^{n-1}\) .

We can find the sum of the first \(k\) terms of a geometric sequence using another trick, which is very different from the one we used for the arithmetic sequence.

Example \(\PageIndex{3}\)

Consider the geometric sequence \(a_n=8\cdot 5^{n-1}\) , that is the sequence:

\[8, 40, 200, 1000, 5000, 25000, 125000, \dots \nonumber \]

We want to add the first \(6\) terms of this sequence. \[8+ 40+ 200+ 1000+ 5000+ 25000=31248 \nonumber \]

In general, it may be much more difficult to simply add the terms as we did above, and we need to use a better general method. For this, we multiply \((1-5)\) to the sum \((8+ 40+ 200+ 1000+ 5000+ 25000)\) and simplify this using the distributive law:

\[\begin{aligned} (1-5)\cdot (8+ 40+ 200+ 1000+ 5000+ 25000) &= 8-40+ 40-200+ 200-1000+ 1000-5000 + 5000-25000+ 25000-125000\\ &= 8-125000 \end{aligned} \nonumber \]

In the second and third lines above, we have what is called a telescopic sum , which can be canceled except for the very first and last terms. Dividing by \((1-5)\) , we obtain:

\[8+ 40+ 200+ 1000+ 5000+ 25000=\dfrac{8-125000}{1-5}=\dfrac{-124992}{-4}=31248 \nonumber \]

The previous example generalizes to the more general setting starting with an arbitrary geometric sequence.

Observation: Geometric series

Let \(\{a_n\}\) be a geometric sequence, whose \(n\) th term is given by the formula \(a_n=a_1\cdot r^{n-1}\) . We furthermore assume that \(r\neq 1\) . Then, the sum \(a_1+a_2+\dots+a_{k-1}+a_k\) is given by:

\[\label{EQU:geometric-series} \boxed{\sum_{i=1}^k a_i =a_1\cdot \dfrac{1-r^k}{1-r}}\]

We multiply \((1-r)\) to the sum \((a_1+a_2+\dots+a_{k-1}+a_k)\) :

\[\begin{aligned} (1-r)\cdot (a_1+a_2+\dots+a_k) &= (1-r)\cdot (a_1 \cdot r^0+a_1\cdot r^1+\dots+a_1\cdot r^{k-1}) \\ &= a_1 \cdot r^0-a_1\cdot r^1+a_1\cdot r^1-a_1\cdot r^2+\dots+a_1 \cdot r^{k-1}-a_1\cdot r^k \\ &= a_1\cdot r^0-a_1\cdot r^k \\&= a_1\cdot (1-r^k)\end{aligned} \nonumber \]

Dividing by \((1-r)\) , we obtain

\[a_1+a_2+\dots+a_k=\dfrac{a_1\cdot (1-r^k)}{(1-r)}=a_1\cdot \dfrac{1-r^k}{1-r} \nonumber \]

This is the formula we wanted to prove.

Example \(\PageIndex{4}\)

Find the value of the geometric series.

• Find the sum \(\sum\limits_{n=1}^6 a_n\) for the geometric sequence \(a_n=10\cdot 3^{n-1}\) .
• Determine the value of the geometric series: \(\sum\limits_{k=1}^{5} \left(-\dfrac{1}{2}\right)^k\)
• Find the sum of the first \(12\) terms of the geometric sequence \[-3, -6, -12, -24, \dots \nonumber\]
• We need to find the sum \(a_1+a_2+a_3+a_4+a_5+a_6\) , and we will do so using the formula provided in equation \(\ref{EQU:geometric-series}\). Since \(a_n=10\cdot 3^{n-1}\) , we have \(a_1=10\) and \(r=3\) , so that

\[\sum_{n=1}^6 a_n = 10\cdot \dfrac{1-3^6}{1-3} = 10\cdot \dfrac{1-729}{1-3}=10\cdot \dfrac{-728}{-2}=10\cdot 364=3640 \nonumber \]

• Again, we use the formula for the geometric series \(\sum_{k=1}^n a_k=a_1\cdot \dfrac{1-r^n}{1-r}\) , since \(a_k=\left(-\dfrac 1 2\right)^k\) is a geometric series. We may calculate the first term \(a_1=-\dfrac 1 2\) , and the common ratio is also \(r=-\dfrac 1 2\) . With this, we obtain:

\[\begin{aligned} \sum_{k=1}^{5} \left(-\dfrac 1 2\right)^k & = \left(-\dfrac 1 2\right) \cdot \dfrac{1-(-\frac 1 2)^5}{1-(-\frac 1 2)}\\& =\left(-\dfrac 1 2\right) \cdot \dfrac{1-\left((-1)^5 \frac {1^5} {2^5}\right)}{1-(-\frac 1 2)}\\&= \left(-\dfrac 1 2\right) \cdot \dfrac{1-(- \frac {1} {32})}{1-(-\frac 1 2)}\\& = \left(-\dfrac 1 2\right) \cdot \dfrac{1+ \frac {1} {32}}{1+\frac 1 2}\\& = \left(-\dfrac 1 2\right) \cdot \dfrac{ \frac {32+1} {32}}{\frac {2+1} 2} \\ &= \left(-\dfrac 1 2\right) \cdot \dfrac{ \frac {33} {32}}{\frac {3} 2} \\&= \left(-\dfrac 1 2\right) \cdot \dfrac {33} {32}\cdot \dfrac 2 3 \\&= -\dfrac 1 2 \cdot \dfrac {11} {16}\\&=-\dfrac {11}{32}\end{aligned} \nonumber \]

• Our first task is to find the formula for the provided geometric series \(-3, -6, -12, -24, \dots\) . The first term is \(a_1=-3\) and the common ratio is \(r=2\) , so that \(a_n=(-3)\cdot 2^{n-1}\) . The sum of the first \(12\) terms of this sequence is again given by equation \(\ref{EQU:geometric-series}\):

\[\begin{aligned} \sum_{i=1}^{12} (-3)\cdot 2^{i-1}&=(-3)\cdot \dfrac{1-2^{12}}{1-2}\\&=(-3)\cdot \dfrac{1-4096}{1-2}\\&=(-3)\cdot \dfrac{-4095}{-1} \\ &= (-3)\cdot 4095\\& = -12285 \end{aligned} \nonumber \]

12.3 Geometric Sequences and Series

Learning objectives.

By the end of this section, you will be able to:

• Determine if a sequence is geometric
• Find the general term (nth term) of a geometric sequence
• Find the sum of the first n n terms of a geometric sequence
• Find the sum of an infinite geometric series
• Apply geometric sequences and series in the real world

Be Prepared 12.7

Before you get started, take this readiness quiz.

Simplify: 24 32 . 24 32 . If you missed this problem, review Example 1.24 .

Be Prepared 12.8

Evaluate: ⓐ 3 4 3 4 ⓑ ( 1 2 ) 4 . ( 1 2 ) 4 . If you missed this problem, review Example 1.19 .

Be Prepared 12.9

If f ( x ) = 4 · 3 x , f ( x ) = 4 · 3 x , find ⓐ f ( 1 ) f ( 1 ) ⓑ f ( 2 ) f ( 2 ) ⓒ f ( 3 ) . f ( 3 ) . If you missed this problem, review Example 3.49 .

Determine if a Sequence is Geometric

We are now ready to look at the second special type of sequence, the geometric sequence.

A sequence is called a geometric sequence if the ratio between consecutive terms is always the same. The ratio between consecutive terms in a geometric sequence is r , the common ratio , where n is greater than or equal to two.

Geometric Sequence

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, a n a n − 1 , a n a n − 1 , is r , the common ratio . n is greater than or equal to two.

Consider these sequences.

Example 12.21

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

ⓐ 4 , 8 , 16 , 32 , 64 , 128 , … 4 , 8 , 16 , 32 , 64 , 128 , …

ⓑ −2 , 6 , −12 , 36 , −72 , 216 , … −2 , 6 , −12 , 36 , −72 , 216 , …

ⓒ 27 , 9 , 3 , 1 , 1 3 , 1 9 , … 27 , 9 , 3 , 1 , 1 3 , 1 9 , …

To determine if the sequence is geometric, we find the ratio of the consecutive terms shown.

Try It 12.41

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

ⓐ 7 , 21 , 63 , 189 , 567 , 1,701 , … 7 , 21 , 63 , 189 , 567 , 1,701 , …

ⓑ 64 , 16 , 4 , 1 , 1 4 , 1 16 , … 64 , 16 , 4 , 1 , 1 4 , 1 16 , …

ⓒ 2 , 4 , 12 , 48 , 240 , 1,440 , … 2 , 4 , 12 , 48 , 240 , 1,440 , …

Try It 12.42

ⓐ −150 , −30 , −15 , −5 , − 5 2 , 0 , … −150 , −30 , −15 , −5 , − 5 2 , 0 , …

ⓑ 5 , 10 , 20 , 40 , 80 , 160 , … 5 , 10 , 20 , 40 , 80 , 160 , …

ⓒ 8 , 4 , 2 , 1 , 1 2 , 1 4 , … 8 , 4 , 2 , 1 , 1 2 , 1 4 , …

If we know the first term, a 1 , a 1 , and the common ratio, r , we can list a finite number of terms of the sequence.

Example 12.22

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

We start with the first term and multiply it by the common ratio. Then we multiply that result by the common ratio to get the next term, and so on.

The sequence is 3 , −6 , 12 , −24 , 48 , … 3 , −6 , 12 , −24 , 48 , …

Try It 12.43

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

Try It 12.44

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

Find the General Term ( n th Term) of a Geometric Sequence

Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence.

Let’s write the first few terms of the sequence where the first term is a 1 a 1 and the common ratio is r . We will then look for a pattern.

As we look for a pattern in the five terms above, we see that each of the terms starts with a 1 . a 1 .

The first term, a 1 , a 1 , is not multiplied by any r . In the second term, the a 1 a 1 is multiplied by r . In the third term, the a 1 a 1 is multiplied by r two times ( r · r r · r or r 2 r 2 ). In the fourth term, the a 1 a 1 is multiplied by r three times ( r · r · r r · r · r or r 3 r 3 ) and in the fifth term, the a 1 a 1 is multiplied by r four times. In each term, the number of times a 1 a 1 is multiplied by r is one less than the number of the term. This leads us to the following

General Term ( n th term) of a Geometric Sequence

The general term of a geometric sequence with first term a 1 a 1 and the common ratio r is

We will use this formula in the next example to find the fourteenth term of a sequence.

Example 12.23

Find the fourteenth term of a sequence where the first term is 64 and the common ratio is r = 1 2 . r = 1 2 .

Try It 12.45

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

Try It 12.46

Find the twelfth term of a sequence where the first term is 256 and the common ratio is r = 1 4 . r = 1 4 .

Sometimes we do not know the common ratio and we must use the given information to find it before we find the requested term.

Example 12.24

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

To find the twelfth term, we use the formula, a n = a 1 r n − 1 , a n = a 1 r n − 1 , and so we need to first determine a 1 a 1 and the common ratio r .

Try It 12.47

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

Try It 12.48

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 n Terms of a Geometric Sequence

We found the sum of both general sequences and arithmetic sequence. We will now do the same for geometric sequences. The sum, S n , S n , of the first n terms of a geometric sequence is written as S n = a 1 + a 2 + a 3 + ... + a n . S n = a 1 + a 2 + a 3 + ... + a n . We can write this sum by starting with the first term, a 1 , a 1 , and keep multiplying by r to get the next term as:

Let’s also multiply both sides of the equation by r .

Next, we subtract these equations. We will see that when we subtract, all but the first term of the top equation and the last term of the bottom equation subtract to zero.

Sum of the First n Terms of a Geometric Series

The sum, S n , S n , of the first n terms of a geometric sequence is

where a 1 a 1 is the first term and r is the common ratio, and r is not equal to one.

We apply this formula in the next example where the first few terms of the sequence are given. Notice the sum of a geometric sequence typically gets very large when the common ratio is greater than one.

Example 12.25

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

To find the sum, we will use the formula S n = a 1 ( 1 − r n ) 1 − r . S n = a 1 ( 1 − r n ) 1 − r . We know a 1 = 7 , a 1 = 7 , r = 2 , r = 2 , and n = 20 . n = 20 .

Try It 12.49

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

Try It 12.50

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

In the next example, we are given the sum in summation notation. While adding all the terms might be possible, most often it is easiest to use the formula to find the sum of the first n terms.

To use the formula, we need r . We can find it by writing out the first few terms of the sequence and find their ratio. Another option is to realize that in summation notation, a sequence is written in the form ∑ i = 1 k a ( r ) i , ∑ i = 1 k a ( r ) i , where r is the common ratio.

Example 12.26

Find the sum: ∑ i = 1 15 2 ( 3 ) i . ∑ i = 1 15 2 ( 3 ) i .

To find the sum, we will use the formula S n = a 1 ( 1 − r n ) 1 − r , S n = a 1 ( 1 − r n ) 1 − r , which requires a 1 a 1 and r . We will write out a few of the terms, so we can get the needed information.

Try It 12.51

Find the sum: ∑ i = 1 15 6 ( 2 ) i . ∑ i = 1 15 6 ( 2 ) i .

Try It 12.52

Find the sum: ∑ i = 1 10 5 ( 2 ) i . ∑ i = 1 10 5 ( 2 ) i .

Find the Sum of an Infinite Geometric Series

If we take a geometric sequence and add the terms, we have a sum that is called a geometric series. An infinite geometric series is an infinite sum whose first term is a 1 a 1 and common ratio is r and is written

Infinite Geometric Series

An infinite geometric series is an infinite sum whose first term is a 1 a 1 and common ratio is r and is written

We know how to find the sum of the first n terms of a geometric series using the formula, S n = a 1 ( 1 − r n ) 1 − r . S n = a 1 ( 1 − r n ) 1 − r . But how do we find the sum of an infinite sum?

Let’s look at the infinite geometric series 3 + 6 + 12 + 24 + 48 + 96 + … . 3 + 6 + 12 + 24 + 48 + 96 + … . Each term gets larger and larger so it makes sense that the sum of the infinite number of terms gets larger. Let’s look at a few partial sums for this series. We see a 1 = 3 a 1 = 3 and r = 2 r = 2

As n gets larger and larger, the sum gets larger and larger. This is true when | r | ≥ 1 | r | ≥ 1 and we call the series divergent. We cannot find a sum of an infinite geometric series when | r | ≥ 1 . | r | ≥ 1 .

Let’s look at an infinite geometric series whose common ratio is a fraction less than one, 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + … 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + … . Here the terms get smaller and smaller as n gets larger. Let’s look at a few finite sums for this series. We see a 1 = 1 2 a 1 = 1 2 and r = 1 2 . r = 1 2 .

Notice the sum gets larger and larger but also gets closer and closer to one. When | r | < 1 , | r | < 1 , the expression r n r n gets smaller and smaller. In this case, we call the series convergent. As n approaches infinity, (gets infinitely large), r n r n gets closer and closer to zero. In our sum formula, we can replace the r n r n with zero and then we get a formula for the sum, S , for an infinite geometric series when | r | < 1 . | r | < 1 .

This formula gives us the sum of the infinite geometric sequence. Notice the S does not have the subscript n as in S n S n as we are not adding a finite number of terms.

Sum of an Infinite Geometric Series

For an infinite geometric series whose first term is a 1 a 1 and common ratio r ,

If | r | < 1 , the sum is If | r | < 1 , the sum is

If | r | ≥ 1 , the infinite geometric series does not have a sum. We say the series diverges. If | r | ≥ 1 , the infinite geometric series does not have a sum. We say the series diverges.

Example 12.27

Find the sum of the infinite geometric series 54 + 18 + 6 + 2 + 2 3 + 2 9 + … 54 + 18 + 6 + 2 + 2 3 + 2 9 + …

To find the sum, we first have to verify that the common ratio | r | < 1 | r | < 1 and then we can use the sum formula S = a 1 1 − r . S = a 1 1 − r .

Try It 12.53

Find the sum of the infinite geometric series 48 + 24 + 12 + 6 + 3 + 3 2 + … 48 + 24 + 12 + 6 + 3 + 3 2 + …

Try It 12.54

Find the sum of the infinite geometric series 64 + 16 + 4 + 1 + 1 4 + 1 16 + … 64 + 16 + 4 + 1 + 1 4 + 1 16 + …

An interesting use of infinite geometric series is to write a repeating decimal as a fraction.

Example 12.28

Write the repeating decimal 0. 5 – 0. 5 – as a fraction.

Try It 12.55

Write the repeating decimal 0. 4 – 0. 4 – as a fraction.

Try It 12.56

Write the repeating decimal 0. 8 – 0. 8 – as a fraction.

Apply Geometric Sequences and Series in the Real World

One application of geometric sequences has to do with consumer spending. If a tax rebate is given to each household, the effect on the economy is many times the amount of the individual rebate.

Example 12.29

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 rebate on the economy?

Every time money goes into the economy, 80% of it is spent and is then in the economy to be spent. Again, 80% of this money is spent in the economy again. This situation continues and so leads us to an infinite geometric series.

Here the first term is 1,000, a 1 = 1000 . a 1 = 1000 . The common ratio is 0.8 , 0.8 , r = 0.8 . r = 0.8 . We can evaluate this sum since 0.8 < 1 . 0.8 < 1 . We use the formula for the sum on an infinite geometric series.

The total effect of the $1,000 received by each household will be a $5,000 growth in the economy.

Try It 12.57

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?

Try It 12.58

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?

We have looked at a compound interest formula where a principal, P , is invested at an interest rate, r , for t years. The new balance, A , is A = P ( 1 + r n ) n t A = P ( 1 + r n ) n t when interest is compounded n times a year. This formula applies when a lump sum was invested upfront and tells us the value after a certain time period.

An annuity is an investment that is a sequence of equal periodic deposits. We will be looking at annuities that pay the interest at the time of the deposits. As we develop the formula for the value of an annuity, we are going to let n = 1 . n = 1 . That means there is one deposit per year.

Suppose P dollars is invested at the end of each year. One year later that deposit is worth P ( 1 + r ) 1 P ( 1 + r ) 1 dollars, and another year later it is worth P ( 1 + r ) 2 P ( 1 + r ) 2 dollars. After t years, it will be worth A = P ( 1 + r ) t A = P ( 1 + r ) t dollars.

After three years, the value of the annuity is

This a sum of the terms of a geometric sequence where the first term is P and the common ratio is 1 + r . 1 + r . We substitute these values into the sum formula. Be careful, we have two different uses of r . The r in the sum formula is the common ratio of the sequence. In this case, that is 1 + r 1 + r where r is the interest rate.

Remember our premise was that one deposit was made at the end of each year.

We can adapt this formula for n deposits made per year and the interest is compounded n times a year.

Value of an Annuity with Interest Compounded n n Times a Year

For a principal, P , invested at the end of a compounding period, with an interest rate, r , which is compounded n times a year, the new balance, A, after t years, is

Example 12.30

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?

To find the Annuity formula, A t = P ( ( 1 + r n ) n t − 1 ) r n , A t = P ( ( 1 + r n ) n t − 1 ) r n , we need to identify P , r , n , and t .

Try It 12.59

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?

Try It 12.60

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?

Access these online resources for additional instruction and practice with sequences.

• Geometric Sequences
• Geometric Series
• Future Value Annuities and Geometric Series
• Application of a Geometric Series: Tax Rebate

Section 12.3 Exercises

Practice makes perfect.

In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.

3 , 12 , 48 , 192 , 768 , 3072 , … 3 , 12 , 48 , 192 , 768 , 3072 , …

2 , 10 , 50 , 250 , 1250 , 6250 , … 2 , 10 , 50 , 250 , 1250 , 6250 , …

72 , 36 , 18 , 9 , 9 2 , 9 4 , … 72 , 36 , 18 , 9 , 9 2 , 9 4 , …

54 , 18 , 6 , 2 , 2 3 , 2 9 , … 54 , 18 , 6 , 2 , 2 3 , 2 9 , …

−3 , 6 , −12 , 24 , −48 , 96 , … −3 , 6 , −12 , 24 , −48 , 96 , …

2 , −6 , 18 , −54 , 162 , −486 , … 2 , −6 , 18 , −54 , 162 , −486 , …

In the following exercises, 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 , 3 2 , … 48 , 24 , 12 , 6 , 3 , 3 2 , …

12 , 6 , 0 , −6 , −12 , −18 , … 12 , 6 , 0 , −6 , −12 , −18 , …

−7 , −2 , 3 , 8 , 13 , 18 , … −7 , −2 , 3 , 8 , 13 , 18 , …

5 , 9 , 13 , 17 , 21 , 25 , … 5 , 9 , 13 , 17 , 21 , 25 , …

1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 64 , … 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 64 , …

4 , 8 , 12 , 24 , 48 , 96 , … 4 , 8 , 12 , 24 , 48 , 96 , …

In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.

a 1 = 4 a 1 = 4 and r = 3 r = 3

a 1 = 9 a 1 = 9 and r = 2 r = 2

a 1 = −4 a 1 = −4 and r = −2 r = −2

a 1 = −5 a 1 = −5 and r = −3 r = −3

a 1 = 27 a 1 = 27 and r = 1 3 r = 1 3

a 1 = 64 a 1 = 64 and r = 1 4 r = 1 4

In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.

Find a 11 a 11 given a 1 = 8 a 1 = 8 and r = 3 . r = 3 .

Find a 13 a 13 given a 1 = 7 a 1 = 7 and r = 2 . r = 2 .

Find a 10 a 10 given a 1 = −6 a 1 = −6 and r = −2 . r = −2 .

Find a 15 a 15 given a 1 = −4 a 1 = −4 and r = −3 . r = −3 .

Find a 10 a 10 given a 1 = 100,000 a 1 = 100,000 and r = 0.1 . r = 0.1 .

Find a 8 a 8 given a 1 = 1,000,000 a 1 = 1,000,000 and r = 0.01 . r = 0.01 .

In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.

Find a 9 a 9 of the sequence, 9 , 18 , 36 , 72 , 144 , 288 , … 9 , 18 , 36 , 72 , 144 , 288 , …

Find a 12 a 12 of the sequence, 5 , 15 , 45 , 135 , 405 , 1215 , … 5 , 15 , 45 , 135 , 405 , 1215 , …

Find a 15 a 15 of the sequence, −486 , 162 , −54 , 18 , −6 , 2 , … −486 , 162 , −54 , 18 , −6 , 2 , …

Find a 16 a 16 of the sequence, 224 , −112 , 56 , −28 , 14 , −7 , … 224 , −112 , 56 , −28 , 14 , −7 , …

Find a 10 a 10 of the sequence, 1 , 0.1 , 0.01 , 0.001 , 0.0001 , 0.00001 , … 1 , 0.1 , 0.01 , 0.001 , 0.0001 , 0.00001 , …

Find a 9 a 9 of the sequence, 1000 , 100 , 10 , 1 , 0.1 , 0.01 , … 1000 , 100 , 10 , 1 , 0.1 , 0.01 , …

Find the Sum of the First n terms of a Geometric Sequence

In the following exercises, find the sum of the first fifteen terms of each geometric sequence.

8 , 24 , 72 , 216 , 648 , 1944 , … 8 , 24 , 72 , 216 , 648 , 1944 , …

7 , 14 , 28 , 56 , 112 , 224 , … 7 , 14 , 28 , 56 , 112 , 224 , …

−6 , 12 , −24 , 48 , −96 , 192 , … −6 , 12 , −24 , 48 , −96 , 192 , …

−4 , 12 , −36 , 108 , −324 , 972 , … −4 , 12 , −36 , 108 , −324 , 972 , …

81 , 27 , 9 , 3 , 1 , 1 3 , … 81 , 27 , 9 , 3 , 1 , 1 3 , …

256 , 64 , 16 , 4 , 1 , 1 4 , 1 16 , … 256 , 64 , 16 , 4 , 1 , 1 4 , 1 16 , …

In the following exercises, find the sum of the geometric sequence.

∑ i = 1 15 ( 2 ) i ∑ i = 1 15 ( 2 ) i

∑ i = 1 10 ( 3 ) i ∑ i = 1 10 ( 3 ) i

∑ i = 1 9 4 ( 2 ) i ∑ i = 1 9 4 ( 2 ) i

∑ i = 1 8 5 ( 3 ) i ∑ i = 1 8 5 ( 3 ) i

∑ i = 1 10 9 ( 1 3 ) i ∑ i = 1 10 9 ( 1 3 ) i

∑ i = 1 15 4 ( 1 2 ) i ∑ i = 1 15 4 ( 1 2 ) i

In the following exercises, find the sum of each infinite geometric series.

1 + 1 3 + 1 9 + 1 27 + 1 81 + 1 243 + 1 729 + … 1 + 1 3 + 1 9 + 1 27 + 1 81 + 1 243 + 1 729 + …

1 + 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + … 1 + 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + …

6 − 2 + 2 3 − 2 9 + 2 27 − 2 81 + … 6 − 2 + 2 3 − 2 9 + 2 27 − 2 81 + …

−4 + 2 − 1 + 1 2 − 1 4 + 1 8 − … −4 + 2 − 1 + 1 2 − 1 4 + 1 8 − …

6 + 12 + 24 + 48 + 96 + 192 + … 6 + 12 + 24 + 48 + 96 + 192 + …

5 + 15 + 45 + 135 + 405 + 1215 + … 5 + 15 + 45 + 135 + 405 + 1215 + …

1,024 + 512 + 256 + 128 + 64 + 32 + … 1,024 + 512 + 256 + 128 + 64 + 32 + …

6,561 + 2187 + 729 + 243 + 81 + 27 + … 6,561 + 2187 + 729 + 243 + 81 + 27 + …

In the following exercises, write each repeating decimal as a fraction.

0. 3 – 0. 3 –

0. 6 – 0. 6 –

0. 7 – 0. 7 –

0. 2 – 0. 2 –

0. 45 — 0. 45 —

0. 27 — 0. 27 —

In the following exercises, 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.

New grandparents decide to invest $ 100 $ 100 per month in an annuity for their grandchild. The account will pay 6 % 6 % interest per year which is compounded monthly (12 times a year). How much will be in the child’s account at their twenty-first birthday?

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

Alice wants to purchase a home in about five years. She is depositing $ 500 $ 500 a month into an annuity that earns 5 % 5 % per year that is compounded monthly (12 times a year). How much will Alice have for her down payment in five years?

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 $ 2,500 a year from her year-end bonus into an annuity. The annuity pays 6.5 % 6.5 % per year and is compounded yearly. How much will she have saved in five years to pursue her master’s degree?

Writing Exercises

In your own words, explain how to determine whether a sequence is geometric.

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

In your own words, explain the difference between a geometric sequence and a geometric series.

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

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction

• Authors: Lynn Marecek, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Intermediate Algebra 2e
• Publication date: May 6, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/intermediate-algebra-2e/pages/12-3-geometric-sequences-and-series

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Module 6: Sequences, Probability, and Counting Theory

Problem set 39: geometric sequences.

1. What is a geometric sequence?

2. How is the common ratio of a geometric sequence found?

3. What is the procedure for determining whether a sequence is geometric?

4. What is the difference between an arithmetic sequence and a geometric sequence?

5. Describe how exponential functions and geometric sequences are similar. How are they different?

For the following exercises, find the common ratio for the geometric sequence.

6. [latex]1,3,9,27,81,..[/latex].

7. [latex]-0.125,0.25,-0.5,1,-2,..[/latex].

8. [latex]-2,-\frac{1}{2},-\frac{1}{8},-\frac{1}{32},-\frac{1}{128},..[/latex].

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

9. [latex]-6,-12,-24,-48,-96,..[/latex].

10. [latex]5,5.2,5.4,5.6,5.8,..[/latex].

11. [latex]-1,\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},..[/latex].

12. [latex]6,8,11,15,20,..[/latex].

13. [latex]0.8,4,20,100,500,..[/latex].

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.

14. [latex]\begin{array}{cc}{a}_{1}=8,& r=0.3\end{array}[/latex]

15. [latex]\begin{array}{cc}{a}_{1}=5,& r=\frac{1}{5}\end{array}[/latex]

For the following exercises, write the first five terms of the geometric sequence, given any two terms.

16. [latex]\begin{array}{cc}{a}_{7}=64,& {a}_{10}\end{array}=512[/latex]

17. [latex]\begin{array}{cc}{a}_{6}=25,& {a}_{8}\end{array}=6.25[/latex]

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.

18. The first term is [latex]2[/latex], and the common ratio is [latex]3[/latex]. Find the 5 th term.

19. The first term is 16 and the common ratio is [latex]-\frac{1}{3}[/latex]. Find the 4 th term.

For the following exercises, find the specified term for the geometric sequence, given the first four terms.

20. [latex]{a}_{n}=\left\{-1,2,-4,8,…\right\}[/latex]. Find [latex]{a}_{12}[/latex].

21. [latex]{a}_{n}=\left\{-2,\frac{2}{3},-\frac{2}{9},\frac{2}{27},…\right\}[/latex]. Find [latex]{a}_{7}[/latex].

For the following exercises, write the first five terms of the geometric sequence.

22. [latex]\begin{array}{cc}{a}_{1}=-486,& {a}_{n}=-\frac{1}{3}\end{array}{a}_{n - 1}[/latex]

23. [latex]\begin{array}{cc}{a}_{1}=7,& {a}_{n}=0.2{a}_{n - 1}\end{array}[/latex]

For the following exercises, write a recursive formula for each geometric sequence.

24. [latex]{a}_{n}=\left\{-1,5,-25,125,…\right\}[/latex]

25. [latex]{a}_{n}=\left\{-32,-16,-8,-4,…\right\}[/latex]

26. [latex]{a}_{n}=\left\{14,56,224,896,…\right\}[/latex]

27. [latex]{a}_{n}=\left\{10,-3,0.9,-0.27,…\right\}[/latex]

28. [latex]{a}_{n}=\left\{0.61,1.83,5.49,16.47,…\right\}[/latex]

29. [latex]{a}_{n}=\left\{\frac{3}{5},\frac{1}{10},\frac{1}{60},\frac{1}{360},…\right\}[/latex]

30. [latex]{a}_{n}=\left\{-2,\frac{4}{3},-\frac{8}{9},\frac{16}{27},…\right\}[/latex]

31. [latex]{a}_{n}=\left\{\frac{1}{512},-\frac{1}{128},\frac{1}{32},-\frac{1}{8},…\right\}[/latex]

32. [latex]{a}_{n}=-4\cdot {5}^{n - 1}[/latex]

33. [latex]{a}_{n}=12\cdot {\left(-\frac{1}{2}\right)}^{n - 1}[/latex]

For the following exercises, write an explicit formula for each geometric sequence.

34. [latex]{a}_{n}=\left\{-2,-4,-8,-16,…\right\}[/latex]

35. [latex]{a}_{n}=\left\{1,3,9,27,…\right\}[/latex]

36. [latex]{a}_{n}=\left\{-4,-12,-36,-108,…\right\}[/latex]

37. [latex]{a}_{n}=\left\{0.8,-4,20,-100,…\right\}[/latex]

38. [latex]{a}_{n}=\left\{-1.25,-5,-20,-80,…\right\}[/latex]

39. [latex]{a}_{n}=\left\{-1,-\frac{4}{5},-\frac{16}{25},-\frac{64}{125},…\right\}[/latex]

40. [latex]{a}_{n}=\left\{2,\frac{1}{3},\frac{1}{18},\frac{1}{108},…\right\}[/latex]

41. [latex]{a}_{n}=\left\{3,-1,\frac{1}{3},-\frac{1}{9},…\right\}[/latex]

For the following exercises, find the specified term for the geometric sequence given.

42. Let [latex]{a}_{1}=4[/latex], [latex]{a}_{n}=-3{a}_{n - 1}[/latex]. Find [latex]{a}_{8}[/latex].

43. Let [latex]{a}_{n}=-{\left(-\frac{1}{3}\right)}^{n - 1}[/latex]. Find [latex]{a}_{12}[/latex].

For the following exercises, find the number of terms in the given finite geometric sequence.

44. [latex]{a}_{n}=\left\{-1,3,-9,…,2187\right\}[/latex]

45. [latex]{a}_{n}=\left\{2,1,\frac{1}{2},…,\frac{1}{1024}\right\}[/latex]

For the following exercises, determine whether the graph shown represents a geometric sequence.

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

48. [latex]\begin{array}{cc}{a}_{1}=1,& r=\frac{1}{2}\end{array}[/latex]

49. [latex]\begin{array}{cc}{a}_{1}=3,& {a}_{n}=2{a}_{n - 1}\end{array}[/latex]

50. [latex]{a}_{n}=27\cdot {0.3}^{n - 1}[/latex]

51. Use recursive formulas to give two examples of geometric sequences whose 3 rd terms are [latex]200[/latex].

52. Use explicit formulas to give two examples of geometric sequences whose 7 th terms are [latex]1024[/latex].

53. Find the 5 th term of the geometric sequence [latex]\left\{b,4b,16b,…\right\}[/latex].

54. Find the 7 th term of the geometric sequence [latex]\left\{64a\left(-b\right),32a\left(-3b\right),16a\left(-9b\right),…\right\}[/latex].

55. At which term does the sequence [latex]\left\{10,12,14.4,17.28,\text{ }…\right\}[/latex] exceed [latex]100?[/latex]

56. At which term does the sequence [latex]\left\{\frac{1}{2187},\frac{1}{729},\frac{1}{243},\frac{1}{81}\text{ }…\right\}[/latex] begin to have integer values?

57. For which term does the geometric sequence [latex]{a}_{{}_{n}}=-36{\left(\frac{2}{3}\right)}^{n - 1}[/latex] first have a non-integer value?

58. Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10 th term.

59. Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8 th term.

60. Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.

• Precalculus. Authored by : Jay Abramson, et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:1/Preface . License : CC BY: Attribution . License Terms : Download for free at: http://cnx.org/contents/[email protected]:1/Preface

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• 1. Multiple Choice Edit 2 minutes 1 pt Find the sum of the infinite geometric series, if it exists: 2 + (16/7) + (128/49) + (1024/343) + ... Does not exist 14 208.32

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