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What Is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers in which each term differs from the previous term by a fixed amount called the common difference. The sequence 3, 7, 11, 15, 19 is arithmetic because each term is 4 more than the one before it. The sequence 100, 95, 90, 85 is also arithmetic — the common difference is negative five, so the sequence decreases at a steady rate.
Arithmetic sequences appear everywhere in everyday life: evenly spaced seat rows in a theatre, monthly savings that grow by a fixed deposit each period, scoring systems that award a consistent number of points per level, and measurement scales with regular intervals are all examples of arithmetic progressions in practice.
How Arithmetic Sequence Formulas Work
Start at a, then add (n − 1) equal steps of size d.
Equivalent shortcut: S_n = n / 2 × (first term + last term).
How to Find the nth Term of an Arithmetic Sequence
The nth term formula is the foundation of all arithmetic sequence work. To find any term by its position number, multiply the position minus one by the common difference and add the result to the first term. The formula is a_n = a + (n − 1) × d.
Consider the sequence 2, 7, 12, 17, 22. The first term is 2 and the common difference is 5. To find the 10th term: 2 + (10 − 1) × 5 = 2 + 45 = 47. The result is 47. The same formula applies regardless of whether the common difference is positive, negative, a whole number, or a decimal.
How to Calculate the Sum of an Arithmetic Sequence
The sum formula uses a shortcut discovered by Carl Friedrich Gauss as a schoolchild. Because the terms of an arithmetic sequence increase evenly, pairing the first term with the last term always gives the same total. Multiply that paired sum by the number of pairs — which is half the number of terms — and the result is the total sum.
Written as a formula: S_n = n / 2 × (first term + last term). An equivalent form is S_n = n / 2 × (2a + (n − 1) × d). Both give the same answer. For the sequence 3, 7, 11, 15, 19, 23, 27 with seven terms: S_7 = 7/2 × (3 + 27) = 3.5 × 30 = 105.
Arithmetic Sequence Formula Reference
| Formula | Expression | Use |
|---|---|---|
| nth term | a_n = a + (n − 1) × d | Find any term by position |
| Sum (standard form) | S_n = n/2 × (2a + (n−1)×d) | Total of first n terms from a and d |
| Sum (first + last) | S_n = n/2 × (a + a_n) | Shortcut when the last term is known |
| Common difference | d = a_n − a_(n−1) | Find d from two consecutive terms |
| Number of terms | n = (a_n − a) / d + 1 | How many terms between a and a_n |
Worked Examples
Example 1 — positive common difference. Find the 10th term and S₁₀ for the sequence 2, 7, 12, 17, ...
The first term is 2 and the common difference is 5. Applying the nth term formula: a₁₀ = 2 + (10 − 1) × 5 = 2 + 45 = 47. For the sum: S₁₀ = 10/2 × (2 × 2 + 9 × 5) = 5 × (4 + 45) = 5 × 49 = 245.
Example 2 — negative common difference. Find the 8th term and S₈ for 50, 47, 44, 41, ...
The first term is 50 and the common difference is negative 3. Applying the nth term formula: a₈ = 50 + (8 − 1) × (−3) = 50 − 21 = 29. For the sum: S₈ = 8/2 × (2 × 50 + 7 × (−3)) = 4 × (100 − 21) = 4 × 79 = 316.
Arithmetic Sequence FAQ
What is an arithmetic sequence? +
An arithmetic sequence is an ordered list of numbers where each term after the first is found by adding a fixed constant — called the common difference — to the previous term. For example, 5, 10, 15, 20 is an arithmetic sequence with a common difference of 5.
How do you find the nth term of an arithmetic sequence? +
Use the formula a_n = a + (n − 1) × d, where a is the first term, d is the common difference, and n is the position number. Multiply (n − 1) by d and add the result to the first term to get the term at position n.
What is the formula for the sum of an arithmetic sequence? +
The sum of the first n terms is S_n = n/2 × (2a + (n−1) × d). An equivalent and often faster shortcut is S_n = n/2 × (first term + last term), which pairs the smallest and largest terms together and multiplies by half the number of terms.
What is the common difference in an arithmetic sequence? +
The common difference is the fixed amount added to each term to produce the next term. It can be positive (increasing sequence), negative (decreasing sequence), zero (constant sequence), or a decimal or fraction. You can find it by subtracting any term from the term that follows it.
Can an arithmetic sequence have a negative common difference? +
Yes. A negative common difference means each term is smaller than the previous one, producing a decreasing sequence. For example, 20, 15, 10, 5, 0, −5 has a common difference of −5. All the same formulas apply — simply substitute the negative value.
What is the difference between an arithmetic sequence and a geometric sequence? +
An arithmetic sequence adds a fixed amount (the common difference) to each term. A geometric sequence multiplies each term by a fixed amount (the common ratio). For example, 2, 5, 8, 11 is arithmetic (add 3 each time), while 2, 6, 18, 54 is geometric (multiply by 3 each time).
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