5.3: Geometric Sequences
In this section, you will:
- Find the common ratio for a geometric sequence.
- List the terms of a geometric sequence.
- Use a recursive formula for a geometric sequence.
- Use an explicit formula for a geometric sequence.
Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.
Finding Common Ratios
The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio . The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.
A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If \( a_1 \) is the initial term of a geometric sequence and \(r\) is the common ratio, the sequence will be\[ \{ a_1 , a_1 r, a_1 r^2 , a_1 r^3 ,\ldots \}. \nonumber \]
Is the sequence geometric? If so, find the common ratio.
- \(1, 2, 4, 8, 16, \ldots \)
- \(48, 12, 4, 2, \ldots \)
- Solution
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Divide each term by the previous term to determine whether a common ratio exists.
- The sequence is geometric because there is a common ratio. The common ratio is 2.
- The sequence is not geometric because there is not a common ratio.
The graph of each sequence is shown in Figure \( \PageIndex{ 1 } \). It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.
Figure \( \PageIndex{ 1 } \)
If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?
No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.
Is the sequence geometric? If so, find the common ratio.
- \(5,10,15,20,\ldots \)
- \(100,20,4, \frac{4}{5} ,\ldots \)
Writing Terms of Geometric Sequences
Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is \( a_1 =−2\) and the common ratio is \(r=4\), we can find subsequent terms by multiplying \(−2 \cdot 4\) to get \(−8\), then multiplying the result \(−8 \cdot 4\) to get \(−32\), and so on.\[ \begin{array}{rcl} a_1 & = & -2 \\[6pt] a_2 & = & (-2) \cdot 4 = -8 \\[6pt] a_3 & = & (-8) \cdot 4 = -32 \\[6pt] a_4 & = & (-32) \cdot 4 = -128 \\[6pt] \end{array} \nonumber \]The first four terms are \( \{–2, –8, –32, –128\} \).
List the first four terms of the geometric sequence with \( a_1 =5\) and \(r=–2\).
- Solution
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Multiply \( a_1 \) by \(−2\) to find \( a_2 \). Repeat the process, using \( a_2 \) to find \( a_3 \), and so on.\[ \begin{array}{rcl} a_1 & = & 5 \\[6pt] a_2 & = & -2 a_1 = (-2) \cdot 5 = -10 \\[6pt] a_3 & = & -2 a_2 = (-2) \cdot (-10) = 20 \\[6pt] a_4 & = & -2 a_3 = (-2) \cdot 20 = -40 \\[6pt] \end{array} \nonumber \]The first four terms are \( \{ 5,–10,20,–40 \} \).
List the first five terms of the geometric sequence with \( a_1 =18\) and \(r= \frac{1}{3} \).
Using Recursive Formulas for Geometric Sequences
A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.
The recursive formula for a geometric sequence with common ratio \(r\) and first term \( a_1 \) is\[ a_n =r a_{n - 1}, \quad n \geq 2. \nonumber \]
Write a recursive formula for the following geometric sequence.\[ \{6, 9, 13.5, 20.25, \ldots \} \nonumber \]
- Solution
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The first term is given as 6. The common ratio can be found by dividing the second term by the first term.\[r = \dfrac{9}{6} =1.5 \nonumber \]Substitute the common ratio into the recursive formula for geometric sequences and define \( a_1 \).\[ \begin{array}{rcl} a_n & = & 1.5 a_{n - 1}, \text{ for }n \geq 2 \\[6pt] a_1 & = & 6 \\[6pt] \end{array} \nonumber \]The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure \( \PageIndex{ 2 } \).
Figure \( \PageIndex{ 2 } \)
Do we have to divide the second term by the first term to find the common ratio?
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.
Write a recursive formula for the following geometric sequence.\[ \left\{2, \dfrac{4}{3}, \dfrac{8}{9}, \dfrac{16}{27}, \ldots \right\} \nonumber \]
Using Explicit Formulas for Geometric Sequences
Because a geometric sequence is an exponential function whose domain is the set of natural numbers and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\[ a_n = a_1 r^{n - 1} \nonumber \]Let’s take a look at the sequence \( \{18, 36, 72, 144, 288, \ldots\} \). This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\[ a_n =18 \cdot 2^{n−1}. \nonumber \]The graph of the sequence is shown in Figure \( \PageIndex{ 3 } \).
Figure \( \PageIndex{ 3 } \)
The \( n^{\text{th}} \) term of a geometric sequence is given by the explicit formula:\[ a_n = a_1 r^{n - 1}. \nonumber \]
Given a geometric sequence with \( a_1 =3\) and \( a_4 =24\), find \( a_2 \).
- Solution
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The sequence can be written in terms of the initial term and the common ratio \(r\).\[3,3r,3 r^2 ,3 r^3 ,\ldots \nonumber \]Find the common ratio using the given fourth term.\[ \begin{array}{rrclcl} & a_n & = & a_1 r^{n - 1} & & \\[6pt] \implies & a_4 & = & 3 r^3 & \quad & \left( \text{substituting }n = 4 \right) \\[6pt] \implies & 24 & = & 3 r^3 & \quad & \left( \text{substituting }a_4 = 24 \right) \\[6pt] \implies & 8 & = & r^3 & \quad & \left( \text{dividing both sides by }3 \right) \\[6pt] \implies & 2 & = & r & \quad & \left( \text{taking the cube root of both sides} \right) \\[6pt] \end{array} \nonumber \]Find the second term by multiplying the first term by the common ratio.\[ a_2 =2 a_1 =2(3) =6 \nonumber \]
Note that the common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.
Given a geometric sequence with \( a_2 =4\) and \( a_3 =32\) , find \( a_6 \).
Write an explicit formula for the \(n^{\text{th}} \) term of the following geometric sequence.\[ \{2, 10, 50, 250, \ldots \} \nonumber \]
- Solution
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The first term is 2. The common ratio can be found by dividing the second term by the first term.\[ \dfrac{10}{2} =5 \nonumber \]The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\[ a_n = 2 \cdot 5^{n−1} \nonumber \]The graph of this sequence in Figure \( \PageIndex{ 4 } \) shows an exponential pattern.
Figure \( \PageIndex{ 4 } \)
Write an explicit formula for the following geometric sequence.\[ \{–1, 3, –9, 27, \ldots \} \nonumber \]
Solving Application Problems with Geometric Sequences
In real-world scenarios involving geometric sequences, we may need to use an initial term of \( a_0 \) instead of \( a_1 \). In these problems, we can alter the explicit formula slightly by using the following formula:\[ a_n = a_0 r^n. \nonumber \]
In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.
- Write a formula for the student population.
- Estimate the student population in 2030.
- Solution
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The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.
Let \(P\) be the student population and \(n\) be the number of years after 2013. Using the explicit formula for a geometric sequence we get\[ P_n = 284 \cdot 1.04^n. \nonumber \]
- We can find the number of years since 2013 by subtracting.\[2030−2013=17 \nonumber \]We are looking for the population after 17 years. We can substitute 17 for \(n\) to estimate the population in 2030.\[ P_{17} = 284 \cdot 1.04^{17} \approx 553. \nonumber \]The student population will be about 553 in 2030.
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A business starts a_new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.
- Write a formula for the number of hits.
- Estimate the number of hits in 5 weeks.
Access these online resources for additional instruction and practice with geometric sequences.