A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given. Show
If you know the n th term of an arithmetic sequence and you know the common difference , d , you can find the ( n + 1 ) th term using the recursive formula a n + 1 = a n + d . Example 1: Find the 9 th term of the arithmetic sequence if the common difference is 7 and the 8 th term is 51 . a 9 = a 8 + d a 9 = 51 + 7 = 58 If you know the n th term and the common ratio , r , of a geometric sequence , you can find the ( n + 1 ) th term using the recursive formula. a n + 1 = a n ⋅ r . Example 2: Write the first four terms of the geometric sequence whose first term is a 1 = 3 and whose common ratio is r = 2 . a 1 = 3 a 2 = a 1 r = 3 ( 2 ) = 6 a 3 = a 2 r = 6 ( 2 ) = 12 a 4 = a 3 r = 12 ( 2 ) = 24 Learning Objectives 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 RatiosThe 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. Definition: GEOMETRIC SEQUENCE 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 \[\{a_1, a_1r,a_1r^2,a_1r^3,...\} .\] How to: Given a set of numbers, determine if they represent a geometric sequence.
Example \(\PageIndex{1}\): Finding Common Ratios Is the sequence geometric? If so, find the common ratio.
Solution Divide each term by the previous term to determine whether a common ratio exists.
Analysis 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 \(\PageIndex{1}\) Q&A 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. Exercise \(\PageIndex{1A}\) Is the sequence geometric? If so, find the common ratio. \(5\), \(10\), \(15\), \(20\),... AnswerThe sequence is not geometric because \(\dfrac{10}{5}≠\dfrac{15}{10}\) Exercise \(\PageIndex{1B}\) Is the sequence geometric? If so, find the common ratio. \(100\), \(20\), \(4\), \(\dfrac{4}{5}\),... AnswerThe sequence is geometric. The common ratio is \(\dfrac{1}{5}\) Writing Terms of Geometric SequencesNow 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 \(−2⋅4\) to get \(−8\) then multiplying the result \(−8⋅4\) to get \(−32\) and so on. \[\begin{align*} a_1 &= −2 \\ a_2 &= (−2⋅4)=−8 \\ a_3 &= (−8⋅4)=−32 \\ a_4 &= (−32⋅4)=128 \end{align*}\] The first four terms are \(\{–2, –8, –32, –128\}\). How to: Given the first term and the common factor, find the first four terms of a geometric sequence.
Example \(\PageIndex{2}\): Writing the Terms of a Geometric Sequence List the first four terms of the geometric sequence with \(a_1=5\) and \(r=–2\). Solution Multiply \(a_1\) by \(−2\) to find \(a_2\). Repeat the process, using \(a_2\) to find \(a_3\), and so on. \[\begin{align*} a_1 &= 5 \\ a_2 &= −2a_1=−10 \\ a_3 &= −2a_2=20 \\ a_4 &= −2a_3=−40 \end{align*}\] The first four terms are \(\{5,–10,20,–40\}\). Exercise \(\PageIndex{2}\) List the first five terms of the geometric sequence with \(a_1=18\) and \(r=\dfrac{1}{3}\). Answer\(\left \{18, 6, 2, \dfrac{2}{3}, \dfrac{2}{9} \right \} \) Using Recursive Formulas for Geometric SequencesA 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. Note: RECURSIVE FORMULA FOR A GEOMETRIC SEQUENCE The recursive formula for a geometric sequence with common ratio r and first term \(a_1\) is \[a_n=ra_{n−1},\;\;\; n≥2\] How to: Given the first several terms of a geometric sequence, write its recursive formula.
Example \(\PageIndex{3}\): Using Recursive Formulas for Geometric Sequences Write a recursive formula for the following geometric sequence. \(\{6, 9, 13.5, 20.25, ...\} \nonumber\) Solution 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 \(a_1\). \[\begin{align*} a_n &= ra_{n−1} \\ a_n &= 1.5a_{n−1} \text{ for }n≥2 \\ a_1 &= 6 \end{align*}\] Analysis 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 \(\PageIndex{2}\) Q&A 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. Exercise \(\PageIndex{3}\) Write a recursive formula for the following geometric sequence. \(\{2, 43, 89, 1627, ...\}\) Answer\(\begin{align*}a_1 &= 2 \\ a_n &= \dfrac{2}{3}a_{n−1} \text{ for }n≥2 \end{align*}\) Using Explicit Formulas for Geometric SequencesBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms. \[a_n=a_1r^{n−1}\] Let’s take a look at the sequence \(\{18, 36, 72, 144, 288, ...\}\). 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 \(a_n=18·2^{n−1}\) The graph of the sequence is shown in Figure \(\PageIndex{3}\). Figure \(\PageIndex{3}\) Note: EXPLICIT FORMULA FOR A GEOMETRIC SEQUENCE The \(n^{th}\) term of a geometric sequence is given by the explicit formula: \[a_n=a_1r^{n−1}\] Example \(\PageIndex{4}\): Writing Terms of Geometric Sequences Using the Explicit Formula Given a geometric sequence with \(a_1=3\) and \(a_4=24\), find \(a_2\). Solution The sequence can be written in terms of the initial term and the common ratio \(r\). \(3\), \(3r\), \(3r^2\), \(3r^3\),... Find the common ratio using the given fourth term. \[\begin{align*} a_n&=a_1r^{n-1} \\ a_4&=3r^3 \qquad \text{Write the fourth term of sequence in terms of }\alpha_1 \text{ and } r \\ 24&=3r^3 \qquad \text{Substitute }24 \text{ for }a_4 \\ 8&=r^3 \qquad \text{Divide} \\ r&=2 \qquad \text{Solve for the common ratio} \end{align*}\] Find the second term by multiplying the first term by the common ratio. \[\begin{align*} a_2 &= 2 \\ a_1 &= 2(3) \\ & = 6 \end{align*}\] Analysis 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. Exercise \(\PageIndex{4}\) Given a geometric sequence with \(a_2=4\) and \(2a_3=32\), find \(a_6\). Answer\(a_6=16,384\) Example \(\PageIndex{5}\): Writing an Explicit Formula for the nth Term of a Geometric Sequence Write an explicit formula for the nth term of the following geometric sequence. \(\{2, 10, 50, 250, ...\}\) Solution The first term is \(2\). The common ratio can be found by dividing the second term by the first term. \(\dfrac{10}{2}=5\) The common ratio is \(5\). Substitute the common ratio and the first term of the sequence into the formula. \[\begin{align*}a_n &= a_1r^{(n−1)} \\ a_n &= 2⋅5^{n−1} \end{align*}\] The graph of this sequence in Figure \(\PageIndex{4}\) shows an exponential pattern. Figure \(\PageIndex{4}\) Exercise \(\PageIndex{5}\) Write an explicit formula for the following geometric sequence. \(\{–1, 3, –9, 27, ...\}\) Answer\(a_n=−{(−3)}^{n−1}\) Solving Application Problems with Geometric SequencesIn real-world scenarios involving arithmetic 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: \(a_n=a_0r^n\) Example \(\PageIndex{6}\): Solving Application Problems with Geometric Sequences 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.
Solution
Exercise \(\PageIndex{6}\) 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.
\(P_n = 293⋅1.026a^n\) Answer bThe number of hits will be about \(333\). Key Equations
Key Concepts
What is a recursive formula?a formula for determining the next term of a sequence from one or more of the preceding terms.
What is the formula in geometric sequence?A geometric sequence is a sequence in which the ratio of any term to the previous term is constant. The explicit formula for a geometric sequence is of the form an = a1r-1, where r is the common ratio.
What is the recursive rule for this geometric sequence 2 6 18 54?The recursive rule for the sequence 2, 6, 18, 54, . . . is an=3an−1 a n = 3 a n − 1 , where an is the nth term of the sequence, and an−1 a n − 1 is the (n-1)st term of the sequence.
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