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

Recognize arithmetic sequences by their constant difference.

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

What is an arithmetic sequence in math?

An arithmetic sequence is a list of numbers where you add the same amount to get from one term to the next. That fixed amount is the common difference. For instance, 7, 11, 15, 19, … adds 4 each time, so the next term is 23. The difference can also be negative, as in 30, 25, 20, 15, …, where each term goes down by 5.

What is the arithmetic sequence formula?

The explicit formula is aₙ = a₁ + (n − 1)d, where a₁ is the first term, d is the common difference, and n is the position of the term. For 6, 10, 14, 18, …, a₁ = 6 and d = 4, so the 20th term is 6 + 19 × 4 = 6 + 76 = 82. The recursive form is aₙ = aₙ₋₁ + d.

What is the common difference in an arithmetic sequence?

The common difference d is the fixed amount added to each term, found by subtracting any term from the next one: d = a₂ − a₁. In 12, 9, 6, 3, …, d = 9 − 12 = −3. Check at least two pairs; if the differences do not match, the pattern is not arithmetic and you may need to test for a common ratio instead.

How do you find the sum of an arithmetic sequence?

Use Sₙ = n(a₁ + aₙ) ÷ 2, which multiplies the number of terms by the average of the first and last terms. To add 3 + 7 + 11 + … + 39, first find n: 39 = 3 + (n − 1) × 4, so n = 10. Then S₁₀ = 10(3 + 39) ÷ 2 = 10 × 42 ÷ 2 = 210.

Are arithmetic sequences linear?

Yes. An arithmetic sequence is a linear function whose inputs are the term numbers 1, 2, 3, and so on. The common difference plays the role of the slope. For example, 5, 8, 11, 14, … can be written aₙ = 3n + 2, which has slope 3. If you plot the terms against their positions, the points fall on a straight line.

Is an arithmetic sequence adding or multiplying?

An arithmetic sequence is built by adding (or subtracting) the same number each time, while a geometric sequence multiplies by the same number. Compare 2, 5, 8, 11, … (add 3) with 2, 6, 18, 54, … (multiply by 3). A quick test is to subtract neighboring terms: a constant difference means arithmetic, while a constant quotient means geometric.

Related Topics

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