How to Solve Arithmetic Sequences? (+FREE Worksheet!)

Arithmetic sequences are number patterns in which each term is found by adding the same fixed number — the common difference — to the previous term. They are among the most common patterns in mathematics and appear in everything from seating arrangements to salary increases. Mastering arithmetic sequences in Algebra 1 builds a strong foundation for understanding linear relationships and series.

What Is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This constant difference is called the common difference (\(\color{blue}{d}\)).

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Example: \(\color{blue}{4, 7, 10, 13, 16, \ldots}\)    Common difference: \(\color{blue}{d = 3}\).

Example: \(\color{blue}{20, 15, 10, 5, \ldots}\)    Common difference: \(\color{blue}{d = -5}\).

Key Formulas for Arithmetic Sequences

Finding the Common Difference

Subtract any term from the term that follows it:

\(\color{blue}{d = a_{n} – a_{n-1}}\)

Explicit (nth-Term) Formula

The nth term of an arithmetic sequence is:

a_n = a₁ + (\(\color{blue}{n – 1}\))d

where \(\color{blue}{a_{1}}\) is the first term, \(\color{blue}{d}\) is the common difference, and \(\color{blue}{n}\) is the term number.

Recursive Formula

Each term is the previous term plus \(\color{blue}{d}\):

\(\color{blue}{a_{n} = a_{n-1} + d}\), with \(\color{blue}{a_{1}}\) given.

Step-by-Step Summary

1. Identify the first term \(\color{blue}{a_{1}}\).
2. Find the common difference: \(\color{blue}{d = a_{2} – a_{1}}\).
3. Write the explicit formula: \(\color{blue}{a_{n} = a_{1} + (n – 1)d}\).
4. Substitute the desired \(\color{blue}{n}\) to find the term.

Watch: Introduction to Arithmetic Sequences (Video Lesson)

Khan Academy introduces arithmetic sequences, identifies the common difference, and explains both explicit and recursive forms:

Arithmetic Sequences – Worked Examples

Example 1: Find the 10th term of \(\color{blue}{4, 7, 10, 13, \ldots}\)

\(\color{blue}{a_{1} = 4}\), \(\color{blue}{d = 3}\).
\(\color{blue}{a_{10} = 4 + (10 – 1)(3) = 4 + 27 = 31}\)

Example 2: Find the 8th term of \(\color{blue}{-5, -1, 3, 7, \ldots}\)

\(\color{blue}{a_{1} = -5}\), \(\color{blue}{d = 4}\).
\(\color{blue}{a_{8} = -5 + (8 – 1)(4) = -5 + 28 = 23}\)

Example 3: Find the 6th term when \(\color{blue}{a_{1} = 100}\) and \(\color{blue}{d = -7}\).

\(\color{blue}{a_{6} = 100 + (6 – 1)(-7) = 100 – 35 = 65}\)

Example 4: Find \(\color{blue}{a_{1}}\) if the 5th term is 23 and \(\color{blue}{d = 3}\).

\(\color{blue}{23 = a_{1} + (5 – 1)(3) = a_{1} + 12 \Rightarrow a_{1} = 11}\)

More Practice: Using Arithmetic Sequence Formulas (Video)

Khan Academy works through several problems using the explicit and recursive formulas to find missing terms and identify sequences:

Exercises for Arithmetic Sequences

1. Find the 7th term of \(\color{blue}{2, 9, 16, 23, \ldots}\)
2. Find the common difference: \(\color{blue}{50, 44, 38, 32, \ldots}\)
3. Write the explicit formula for \(\color{blue}{1, 4, 7, 10, \ldots}\)
4. Find \(\color{blue}{a_{12}}\) when \(\color{blue}{a_{1} = -3}\) and \(\color{blue}{d = 5}\).
5. Find \(\color{blue}{a_{1}}\) if \(\color{blue}{a_{6} = 41}\) and \(\color{blue}{d = 6}\).

Answers

1. \(\color{blue}{a_{7} = 2 + 6 \times 7 = 44}\)
2. \(\color{blue}{d = -6}\)
3. \(\color{blue}{a_{n} = 1 + (n – 1)(3) = 3n – 2}\)
4. \(\color{blue}{a_{12} = -3 + 11 \times 5 = 52}\)
5. \(\color{blue}{41 = a_{1} + 5 \times 6 \Rightarrow a_{1} = 11}\)

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Free Arithmetic Sequences Worksheet

Ready to practice on your own? Download our free Arithmetic Sequences worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Arithmetic Sequences before a quiz or test.

Download Arithmetic Sequences as Linear Functions Worksheet

Frequently Asked Questions

How is an arithmetic sequence related to a linear equation?

Both involve a constant rate of change. The explicit formula \(\color{blue}{a_{n} = a_{1} + (n – 1)d}\) is a linear function of \(\color{blue}{n}\), similar to \(\color{blue}{y = \text{ mx } + b}\) where the slope is \(\color{blue}{d}\).

Can the common difference be negative?

Yes. A negative common difference means the sequence is decreasing. For example, \(\color{blue}{d = -5}\) gives \(\color{blue}{20, 15, 10, 5, 0, -5, \ldots}\)

What is the difference between an arithmetic sequence and a geometric sequence?

An arithmetic sequence uses addition (constant difference), while a geometric sequence uses multiplication (constant ratio). Arithmetic sequences grow linearly; geometric sequences grow exponentially.

Related Topics

• Geometric Sequences
• Writing Linear Equations
• Finding Slope
• Unit Rates and Rates