A recursive rule gives the first term or terms of a sequence and describes how each term is related to the preceding term(s) with a recursive equation. For example, arithmetic and geometric sequences can be described recursively.
Example arithmetic sequence | Example geometric sequence |
Recursive equation | |
$\begin{gathered} a_1=3 , \ \ \ a_n \! = \! a_{n-1}\! + \! 2 \end{gathered}$ | $\begin{gathered} a_1=5 , \ \ \ a_n \! = \! a_{n-1}\! \times \! 2 \end{gathered}$ |
The recursive rule for an arithmetic sequence can be expressed as $a_n=a_{n-1}+d$ where $a_{n-1}$ and $a_n$ are consecutive terms and $d$ is the common difference. To write the rule, $d$ must be found. Consider the arithmetic sequence $5,\, 11,\, 17,\, 23,\, \ldots$ To find $d,$ we can subtract any term from the term that comes after it. Let's use $a_2={\color{#0000FF}{11}}$ and $a_1={\color{#009600}{5}}.$ $d=a_2-a_1={\color{#0000FF}{11}}-{\color{#009600}{5}}=6$ Using $d=6,$ the recursive rule can be written as $a_n=a_{n-1}+6.$ As it's written, this rule describes any sequence with a common difference of $6.$ For example, it can describe $\text{-} 45,\,\text{-} 39,\,\text{-} 33,\,\text{-}27, \ldots$ as well as $111, 117, 123, 129, 135, \ldots$ To ensure the recursive rule defines the given sequence, it is necessary to also give the first term, $a_1.$ Thus, the recursive rule for the given sequence is $\begin{aligned} &a_1=5 \\ &a_n=a_{n-1}+6. \end{aligned}$ Now that the recursive rule is known, it can be used to find any term, provided that the previous term is given. For example, since $a_4=23,$ $a_5$ can be found. $\begin{aligned} a_n&=a_{n-1}+6 \\ a_5&=a_4+6 \\ a_5&= 23+ 6 \\ a_5&=29 \end{aligned}$
Thus, the $5\text{th}$ term in the sequence is $29.$Bonnie recently purchased a used car. Each weekend Bonnie records the odometer of her car. The first four weekends she noted the following miles. $\begin{aligned} 26\,247 && 26\,311 \\ 26\,375 && 26\,439 \end{aligned}$
Create a recursive rule that describes the sequence. Then use the recursive rule to find the odometer's reading the following two weeks, assuming that Bonnie drives exactly the same distance every week.
Before we write the recursive rule for the sequence, let's ensure it's arithmetic.
Notice that Bonnie drives $64$ miles each week. Therefore, the odometer readings form an arithmetic sequence with a common difference of $d=64.$ Using $d$ and the given information we can write the recursive rule as follows. $\begin{aligned} a_1&=26\,247 \\ a_n&=a_{n-1}+64 \end{aligned}$ We can use the recursive rule to find the odometer readings for the next two weeks, $a_5$ and $a_6.$ We'll use $a_4=26\,439$ to find $a_5.$ $\begin{aligned} a_n&=a_{n-1}+64 \\ a_5&=a_4+64 \\ a_5&=26\,439+64 \\ a_5&=26\,503 \end{aligned}$ Similarly, we can add $64$ to $a_5$ to find the odometer reading of the next weekend. $a_6=a_5+64=26\,503+64=26\,567$ Thus, after the fifth weekend, the odometer will read $26\,503$ miles and after the sixth weekend, it will read $26\,567$ miles.
An arithmetic sequence is given by the following explicit rule. Translate it into a recursive rule. $a_n=4+(n-1)6.$
A recursive rule for an arithmetic sequence is written as $\begin{aligned} a_1&= \ldots \\ a_n&=a_{n-1}+d, \end{aligned}$
where $a_1$ is the first term and $d$ is the common difference. Since an explicit rule for an arithmetic sequence is written as $a_n=a_1+(n-1)d,$ it can be seen that $a_1=4$ and $d=6.$ Thus, we can write the recursive rule for this sequence as $\begin{aligned} a_1&=4\\ a_n&=a_{n-1}+6. \end{aligned}$
An arithmetic sequence is defined by the following recursive rule. Write the explicit rule for the same sequence. $\begin{array}{l}a_1=\text{-} 5 \\ a_n=a_{n-1}+7 \end{array}$
The explicit rule of an arithmetic sequence is written as $a_n=a_1+(n-1)d,$ where $a_1$ is the first term and $d$ is the common difference. To write an explicit rule, both $a_1$ and $d$ are needed. From the recursive rule $\begin{aligned} a_1&=\text{-} 5\\ a_n&=a_{n-1}+7 \end{aligned}$ it can be seen that $a_1=\text{-} 5$ and $d=7.$ Thus, the explicit rule can be written and simplified as follows. $a_n=\text{-} 5+(n-1)7 \quad \Rightarrow \quad a_n=\text{-}12 +7n$