9.3: Geometric Sequences and Series

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Identify the common ratio of a geometric sequence.
Find a formula for the general term of a geometric sequence.
Calculate the $n$th partial sum of a geometric sequence.
Calculate the sum of an infinite geometric series when it exists.

Geometric Sequences

A geometric sequence 18 , or geometric progression 19 , is a sequence of numbers where each successive number is the product of the previous number and some constant $r$.

And because $\frac>>=r$, the constant factor $r$ is called the common ratio 20 . For example, the following is a geometric sequence,

Here $a_ = 9$ and the ratio between any two successive terms is $3$. We can construct the general term $a_=3 a_$ where,

\[\begin a_ &=9 \\ a_ &=3 a_=3(9)=27 \\ a_ &=3 a_=3(27)=81 \\ a_ &=3 a_=3(81)=243 \\ a_ &=3 a_=3(243)=729 \\ & \vdots \end\]

In general, given the first term $a_$ and the common ratio $r$ of a geometric sequence we can write the following:

$\begin a_ &=r a_ \\ a_ &=r a_=r\left(a_ r\right)=a_ r^ \\ a_ &=r a_=r\left(a_ r^\right)=a_ r^ \\ a_ &=r a_=r\left(a_ r^\right)=a_ r^ \\ & \vdots \end$

From this we see that any geometric sequence can be written in terms of its first element, its common ratio, and the index as follows:

In fact, any general term that is exponential in $n$ is a geometric sequence.

Find an equation for the general term of the given geometric sequence and use it to calculate its $10^$ term: $3, 6, 12, 24, 48…$

Solution

Begin by finding the common ratio,

Note that the ratio between any two successive terms is $2$. The sequence is indeed a geometric progression where $a_ = 3$ and $r = 2$.

Therefore, we can write the general term $a_=3(2)^$ and the $10^$ term can be calculated as follows:

Answer:

The terms between given terms of a geometric sequence are called geometric means 21 .

Find all terms between $a_ = −5$ and $a_ = −135$ of a geometric sequence. In other words, find all geometric means between the $1^$ and $4^$ terms.

Solution

Begin by finding the common ratio $r$. In this case, we are given the first and fourth terms:

Substitute $a_ = −5$ and $a_ = −135$ into the above equation and then solve for $r$.

Next use the first term $a_ = −5$ and the common ratio $r = 3$ to find an equation for the $n$th term of the sequence.

Now we can use $a_=-5(3)^$ where $n$ is a positive integer to determine the missing terms.

Answer:

The first term of a geometric sequence may not be given.

Find the general term of a geometric sequence where $a_ = −2$ and $a_=\frac$.

Solution

To determine a formula for the general term we need $a_$ and $r$. A nonlinear system with these as variables can be formed using the given information and $a_=a_ r^ :$:

Solve for $a_$ in the first equation,

Substitute $a_ = \frac$ into the second equation and solve for $r$.

Back substitute to find $a_$:

Therefore, $a_ = 10$ and $r = −\frac$.

Answer:

Find an equation for the general term of the given geometric sequence and use it to calculate its $6^$ term: $2, \frac,\frac, …$

Answer

Geometric Series

A geometric series 22 is the sum of the terms of a geometric sequence. For example, the sum of the first $5$ terms of the geometric sequence defined by $a_=3^$ follows:

Adding $5$ positive integers is manageable. However, the task of adding a large number of terms is not. Therefore, we next develop a formula that can be used to calculate the sum of the first $n$ terms of any geometric sequence. In general,

Multiplying both sides by $r$ we can write,

Subtracting these two equations we then obtain,

Assuming $r ≠ 1$ dividing both sides by $(1 − r)$ leads us to the formula for the $n$th partial sum of a geometric sequence 23 :

In other words, the $n$th partial sum of any geometric sequence can be calculated using the first term and the common ratio. For example, to calculate the sum of the first $15$ terms of the geometric sequence defined by $a_=3^$, use the formula with $a_ = 9$ and $r = 3$.

Find the sum of the first 10 terms of the given sequence: $4, −8, 16, −32, 64,… $

Solution

Determine whether or not there is a common ratio between the given terms.

Note that the ratio between any two successive terms is $−2$; hence, the given sequence is a geometric sequence. Use $r = −2$ and the fact that $a_ = 4$ to calculate the sum of the first $10$ terms,

Answer:

Solution

In this case, we are asked to find the sum of the first $6$ terms of a geometric sequence with general term $a_ = 2(−5)^$. Use this to determine the $1^$ term and the common ratio $r$:

To show that there is a common ratio we can use successive terms in general as follows:

Use $a_ = −10$ and $r = −5$ to calculate the $6^$ partial sum.

Answer:

Find the sum of the first 9 terms of the given sequence: $-2,1,-1 / 2, \dots$

Answer

If the common ratio r of an infinite geometric sequence is a fraction where $|r| < 1$ (that is $−1 < r < 1$), then the factor $(1 − r^)$ found in the formula for the $n$th partial sum tends toward $1$ as $n$ increases. For example, if $r = \frac$ and $n = 2, 4, 6$ we have,

Here we can see that this factor gets closer and closer to 1 for increasingly larger values of $n$. This illustrates the idea of a limit, an important concept used extensively in higher-level mathematics, which is expressed using the following notation:

This is read, “the limit of $(1 − r^)$ as $n$ approaches infinity equals $1$.” While this gives a preview of what is to come in your continuing study of mathematics, at this point we are concerned with developing a formula for special infinite geometric series. Consider the $n$th partial sum of any geometric sequence,

Therefore, a convergent geometric series 24 is an infinite geometric series where $|r| < 1$; its sum can be calculated using the formula:

Find the sum of the infinite geometric series: $\frac+\frac+\frac+\frac+\frac+\dots$

Solution

Determine the common ratio, Since the common ratio $r = \frac$ is a fraction between $−1$ and $1$, this is a convergent geometric series. Use the first term $a_ = \frac$ and the common ratio to calculate its sum

Answer:

In the case of an infinite geometric series where $|r| ≥ 1$, the series diverges and we say that there is no sum. For example, if $a_ = (5)^$ then $r = 5$ and we have

We can see that this sum grows without bound and has no sum.

Find the sum of the infinite geometric series: $\sum_^<\infty>-2\left(\frac\right)^$

Answer

A repeating decimal can be written as an infinite geometric series whose common ratio is a power of $1/10$. Therefore, the formula for a convergent geometric series can be used to convert a repeating decimal into a fraction.

Write as a fraction: $1.181818… $

Solution

Begin by identifying the repeating digits to the right of the decimal and rewrite it as a geometric progression.

In this form we can determine the common ratio,

Note that the ratio between any two successive terms is $\frac$. Use this and the fact that $a_ = \frac$ to calculate the infinite sum:

Therefore, $0.181818… = \frac$ and we have,

Answer:

A certain ball bounces back to two-thirds of the height it fell from. If this ball is initially dropped from $27$ feet, approximate the total distance the ball travels.

Solution

We can calculate the height of each successive bounce:

$26d216617d1151893f08a09878633d19.png$

The total distance that the ball travels is the sum of the distances the ball is falling and the distances the ball is rising. The distances the ball falls forms a geometric series,

where $a_ = 27$ and $r = \frac$. Because $r$ is a fraction between $−1$ and $1$, this sum can be calculated as follows:

Therefore, the ball is falling a total distance of $81$ feet. The distances the ball rises forms a geometric series,

where $a_ = 18$ and $r = \frac$. Calculate this sum in a similar manner:

Therefore, the ball is rising a total distance of $54$ feet. Approximate the total distance traveled by adding the total rising and falling distances:

Answer:

Key Takeaways

A geometric sequence is a sequence where the ratio $r$ between successive terms is constant.
The general term of a geometric sequence can be written in terms of its first term $a_$, common ratio $r$, and index $n$ as follows: $a_ = a_ r^$.
A geometric series is the sum of the terms of a geometric sequence.
The $n$th partial sum of a geometric sequence can be calculated using the first term $a_$ and common ratio $r$ as follows: $S_=\frac$.
The infinite sum of a geometric sequence can be calculated if the common ratio is a fraction between $−1$ and $1$ (that is $|r| < 1$) as follows: $S_<\infty>=\frac$. If $|r| ≥ 1$, then no sum exists.

Write the first $5$ terms of the geometric sequence given its first term and common ratio. Find a formula for its general term.

$a_=1 ; r=5$
$a_=1 ; r=3$
$a_=2 ; r=3$
$a_=5 ; r=4$
$a_=2 ; r=-3$
$a_=6 ; r=-2$
$a_=3 ; r=\frac$
$a_=6 ; r=\frac$
$a_=1.2 ; r=0.6$
$a_=-0.6 ; r=-3$

Given the geometric sequence, find a formula for the general term and use it to determine the $5^$ term in the sequence.

$7,28,112, \dots$
$-2,-10,-50, \dots$
$2, \frac, \frac, \ldots$
$1, \frac, \frac, \ldots$
$8,4,2, \dots$
$6,2, \frac, \ldots$
$-1, \frac,-\frac, \ldots$
$2,-\frac, \frac, \ldots$
$\frac,-2,12, \dots$
$\frac,-2,10, \dots$
$-3.6,-4.32,-5.184, \dots$
$0.8,-2.08,5.408, \dots$
Find the general term and use it to determine the $20^$ term in the sequence: $1, \frac, \frac, \ldots$
Find the general term and use it to determine the $20^$ term in the sequence: $2,-6 x, 18 x^ \ldots$
The number of cells in a culture of a certain bacteria doubles every $4$ hours. If $200$ cells are initially present, write a sequence that shows the population of cells after every $n$th $4$-hour period for one day. Write a formula that gives the number of cells after any $4$-hour period.
A certain ball bounces back at one-half of the height it fell from. If this ball is initially dropped from $12$ feet, find a formula that gives the height of the ball on the $n$th bounce and use it to find the height of the ball on the $6^$ bounce.
Given a geometric sequence defined by the recurrence relation $a_ = 4a_$ where $a_ = 2$ and $n > 1$, find an equation that gives the general term in terms of $a_$ and the common ratio $r$.
Given the geometric sequence defined by the recurrence relation $a_ = 6a_$ where $a_ = \frac$ and $n > 1$, find an equation that gives the general term in terms of $a_$ and the common ratio $r$.

15. $400$ cells; $800$ cells; $1,600$ cells; $3,200$ cells; $6,400$ cells; $12,800$ cells; $p_ = 400(2)^$ cells

Given the terms of a geometric sequence, find a formula for the general term.

$a_=-3$ and $a_=-96$
$a_=5$ and $a_=-40$
$a_=-2$ and $a_=-\frac$
$a_=\frac$ and $a_=-\frac$
$a_=18$ and $a_=486$
$a_=10$ and $a_=320$
$a_=-2$ and $a_=64$
$a_=-\frac$ and $a_=\frac$
$a_=153.6$ and $a_=9,830.4$
$a_=-2.4 \times 10^$ and $a_=-7.68 \times 10^$

Find all geometric means between the given terms.

$a_=2$ and $a_=250$
$a_=\frac$ and $a_=-\frac$
$a_=-20$ and $a_=-20,000$
$a_=49$ and $a_=-16,807$

Calculate the indicated sum.

$a_=2^ ; S_$
$a_=(-2)^ ; S_$
$a_=\left(\frac\right)^ ; S_$
$a_=\left(\frac\right)^ ; S_$
$a_=5(-3)^ ; S_$
$a_=-7(-4)^ ; S_$
$a_=2\left(-\frac\right)^ ; S_$
$a_=\frac(2)^ ; S_$
$\sum_^ 5^$
$\sum_^(-4)^$
$\sum_^ 2^$
$\sum_^ 2^$
$\sum_^-2(3)^$
$\sum_^ 5(-2)^$
$\sum_^ 2\left(\frac\right)^$
$\sum_^-3\left(\frac\right)^$
$a_=\left(\frac\right)^ ; S_<\infty>$
$a_=\left(\frac\right)^ ; S_<\infty>$
$a_=2\left(-\frac\right)^ ; S_<\infty>$
$a_=3\left(-\frac\right)^ ; S_<\infty>$
$a_=-2\left(\frac\right)^ ; S_<\infty>$
$a_=-\frac\left(-\frac\right)^ ; S_<\infty>$
$\sum_^ <\infty>2\left(\frac\right)^$
$\sum_^<\infty>\left(\frac\right)^$
$\sum_^ <\infty>3(2)^$
$\sum_^<\infty>-\frac(3)^$
$\sum_^ <\infty>\frac\left(-\frac\right)^$
$\sum_^ <\infty>\frac\left(-\frac\right)^$

Write as a mixed number.

$7b6e86da33720f5b366f2e4164b09d0d.png$

$1.222…$
$5.777 …$
$2.252525…$
$3.272727…$
$1.999…$
$1.090909…$
Suppose you agreed to work for pennies a day for $30$ days. You will earn $1$ penny on the first day, $2$ pennies the second day, $4$ pennies the third day, and so on. How many total pennies will you have earned at the end of the $30$ day period? What is the dollar amount?
An initial roulette wager of $$100$ is placed (on red) and lost. To make up the difference, the player doubles the bet and places a $$200$ wager and loses. Again, to make up the difference, the player doubles the wager to $$400$ and loses. If the player continues doubling his bet in this manner and loses $7$ times in a row, how much will he have lost in total?
A certain ball bounces back to one-half of the height it fell from. If this ball is initially dropped from $12$ feet, approximate the total distance the ball travels.
A golf ball bounces back off of a cement sidewalk three-quarters of the height it fell from. If the ball is initially dropped from $8$ meters, approximate the total distance the ball travels.
A structured settlement yields an amount in dollars each year, represented by $n$, according to the formula $p_ = 6,000(0.80)^$. What is the total amount gained from the settlement after $10$ years?
Beginning with a square, where each side measures $1$ unit, inscribe another square by connecting the midpoints of each side. Continue inscribing squares in this manner indefinitely, as pictured:

Find the sum of the area of all squares in the figure. (Hint: Begin by finding the sequence formed using the areas of each square.)

Answer

7. $1,073,741,823$ pennies; $\$ 10,737,418.23$

Categorize the sequence as arithmetic, geometric, or neither. Give the common difference or ratio, if it exists.

$-12,24,-48, \dots$
$-7,-5,-3, \dots$
$-3,-11,-19, \dots$
$4,9,16, \dots$
$2, \frac, \frac, \dots$
$\frac, \frac, \frac, \dots$
$\frac,-\frac,-\frac, \ldots$
$\frac, \frac, \frac, \dots$
$\frac, \frac, \frac\dots$
$-\frac,-\frac,-\frac, \dots$
$1.26,0.252,0.0504, \dots$
$0.02,0.08,0.18, \dots$
$1,-1,1,-1, \ldots$
$0,0,0, \dots$

1. Geometric; $r = −2$

3. Arithmetic; $d = −8$

7. Arithmetic; $d = −\frac$

11. Geometric; $r = 0.2$

13. Geometric; $r = −1$

Categorize the sequence as arithmetic or geometric, and then calculate the indicated sum.

1. Geometric; $292,968$

3. Arithmetic; $210$

Calculate the indicated sum.

Use the techniques found in this section to explain why $0.999… = 1$.
Construct a geometric sequence where $r = 1$. Explore the $n$th partial sum of such a sequence. What conclusions can we make?

1. Answer may vary

Footnotes

18 A sequence of numbers where each successive number is the product of the previous number and some constant $r$.

19 Used when referring to a geometric sequence.

20 The constant $r$ that is obtained from dividing any two successive terms of a geometric sequence; $\frac>>=r$.

21 The terms between given terms of a geometric sequence.

22 The sum of the terms of a geometric sequence.

23 The sum of the first n terms of a geometric sequence, given by the formula: $S_=\frac\left(1-r^\right)> , r\neq 1$.

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9.3: Geometric Sequences and Series

Geometric Sequences

Geometric Series

Key Takeaways

Footnotes

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