MA001: College Algebra

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
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