Use a Formula for an Arithmetic Sequence

Some arithmetic sequences are defined in terms of the previous term using a recursive formula. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.

$a_n=a_{n−1}+d \quad n \geq 2$

Recursive Formula for an Arithmetic Sequence

The recursive formula for an arithmetic sequence with common difference $d$ is:

$a_n=a_{n−1}+d \quad n \geq 2$

How To

Given an arithmetic sequence, write its recursive formula.

1. Subtract any term from the subsequent term to find the common difference.
   
2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.
   

Example 4

Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic sequence.

$\{−18, −7, 4, 15, 26, …\}$

Solution

The first term is given as $−18$. The common difference can be found by subtracting the first term from the second term.

$d=−7−(−18)=11$

Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.

$\begin{array}{ll} a_1 = -18 \\ a_n = a_{n-1} + 11, \text { for } n \geq 2 \end{array}$

Analysis

We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure 3. The growth pattern of the sequence shows the constant difference of 11 units.

Figure 3

Q&A

Do we have to subtract the first term from the second term to find the common difference?

No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.

Try It #5

Write a recursive formula for the arithmetic sequence.

$\{25,  37,  49,  61,  …\}$

Source: Rice University, https://openstax.org/books/college-algebra/pages/9-2-arithmetic-sequences
This work is licensed under a Creative Commons Attribution 4.0 License.

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.

$a_n=a_1+d(n−1)$

To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.

The common difference is $−50$, so the sequence represents a linear function with a slope of $−50$. To find the $y$ -intercept, we subtract $−50$ from $200:200−(−50)=200+50=250$. You can also find the $y$ -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in Figure 4.

Figure 4

Recall the slope-intercept form of a line is $y=mx+b$. When dealing with sequences, we use $a_n$ in place of $y$ and $n$ in place of $x$. If we know the slope and vertical intercept of the function, we can substitute them for $m$ and $b$ in the slope-intercept form of a line. Substituting $−50$ for the slope and $250$ for the vertical intercept, we get the following equation:

$a_n=−50n+250$

We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is $a_n=200−50(n−1)$, which simplifies to $a_n=−50n+250$.

Explicit Formula for an Arithmetic Sequence

An explicit formula for the $nth$ term of an arithmetic sequence is given by

$a_n=a_1+d(n−1)$

How To

Given the first several terms for an arithmetic sequence, write an explicit formula.

1. Find the common difference, $a_2−a_1$.
   
2. Substitute the common difference and the first term into $a_n=a_1+d(n−1)$.
   

Example 5

Writing the $nth$ Term Explicit Formula for an Arithmetic Sequence

Write an explicit formula for the arithmetic sequence.

$\{2, 12, 22, 32, 42, …\}$

Solution

$\begin{array}{ll} d &= a_2 - a_1 \\ &= 12 - 2 \\ &=10 \end{array}$

$\begin{array}{ll} a_n = 2+10(n-1) \\ a_n = 10n-8 \end{array}$

Analysis

The graph of this sequence, represented in Figure 5, shows a slope of 10 and a vertical intercept of $−8$.

Figure 5

Try It #6

Write an explicit formula for the following arithmetic sequence.

$\{50,47,44,41,…\}$

Finding the Number of Terms in a Finite Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

How To

Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

1. Find the common difference $d$.
   
2. Substitute the common difference and the first term into $a_n=a_1+d(n–1)$.
   
3. Substitute the last term for $a_n$ and solve for $n$.
   

Example 6

Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence.

$\{8, 1, –6, ..., –41\}$

Solution

The common difference can be found by subtracting the first term from the second term.

$1−8=−7$

The common difference is$−7$. Substitute the common difference and the initial term of the sequence into the $nth$ term formula and simplify.

$\begin{array}{ll} a_n = a_1+d(n-1) \\ a_n = 8+(-7)(n-1) \\ a_n= 15 - 7n \end{array}$

$\begin{array}{ll} -41 &= 15 -7n \\ 8 &= n \end{array}$

Solving Application Problems with Arithmetic Sequences

In many application problems, it often makes sense to use an initial term of $a_0$ instead of $a_1$. In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:

$a_n=a_0+dn$

Example 7

Solving Application Problems with Arithmetic Sequences

A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.

Solution

Try It #8

A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?