Arithmetic Sequence Calculator
Enter the first term (a₁), common difference (d), and the term number (n) to find the nth term, the sum of the first n terms, and a list of the first 10 terms.
What is an Arithmetic Sequence?
An arithmetic sequence (also called an arithmetic progression or AP) is an ordered list of numbers in which each term after the first is obtained by adding a fixed constant called the common difference (d) to the preceding term. For example, 3, 7, 11, 15, 19 … is an arithmetic sequence with a first term (a₁) of 3 and a common difference of 4. The sequence is called "arithmetic" because each step involves the same arithmetic operation — addition of a constant.
Arithmetic sequences are linear in nature: when plotted on a graph with the term number (n) on the x-axis and the term value on the y-axis, the points fall exactly on a straight line with slope equal to the common difference. This makes them among the simplest and most predictable mathematical sequences, and they arise naturally in situations with constant rates of change — regular savings deposits, hourly wages, evenly spaced physical measurements, and many scheduling problems.
Two core formulas govern arithmetic sequences: the nth term formula (to find any specific term without listing all previous terms) and the sum formula (to efficiently total any number of consecutive terms). The sum formula, attributed to the young Gauss, recognises that pairing terms from opposite ends of a finite sequence always produces the same total — making the sum equal to the average of the first and last terms, multiplied by the number of terms.
Arithmetic Sequence Formulas
How the Arithmetic Sequence Calculator Works
Formula, assumptions, and calculation steps for this math tool.
Methodology
Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.
Calculation Steps
- Read the values and operation selected.
- Normalize signs, decimals, fractions, or units if needed.
- Apply the mathematical rule or formula.
- Format the answer and any intermediate values for checking.
Assumptions and Limits
- Inputs must be within the supported domain of the operation.
- Decimal answers may be rounded for readability.
- Symbolic simplification is limited to the calculator scope.
Frequently Asked Questions
An arithmetic sequence (or arithmetic progression) is a list of numbers where consecutive terms differ by a constant value called the common difference (d). Example: 2, 5, 8, 11… with d = 3.
Use the formula aₙ = a₁ + (n−1)d, where a₁ is the first term, d is the common difference, and n is the position.
Sₙ = n/2 × (2a₁ + (n−1)d). This formula sums the first n terms. It can also be written as Sₙ = n/2 × (first term + last term).
The common difference (d) is the fixed amount added (or subtracted) to get from one term to the next. For 3, 7, 11, 15 the common difference is 4.
Yes. A negative common difference creates a decreasing sequence. For example, 20, 15, 10, 5, 0, −5… has d = −5.
Real-World Applications of Arithmetic Sequences
Common Arithmetic Sequence Mistakes
Arithmetic vs Geometric vs Fibonacci Sequences
| Type | Rule | Example | Growth |
|---|---|---|---|
| Arithmetic | Add constant d each term | 2, 5, 8, 11 (d=3) | Linear |
| Geometric | Multiply by constant r each term | 2, 6, 18, 54 (r=3) | Exponential |
| Fibonacci | Each term = sum of previous two | 1, 1, 2, 3, 5, 8, 13 | Approx. exponential |
| Harmonic | Reciprocals form arithmetic seq | 1, ½, ⅓, ¼, ⅕ | Decreasing, divergent sum |
References
- Stewart J. Calculus: Early Transcendentals. 8th ed. Cengage, 2015. (Ch. 11: Sequences and Series)
- Khan Academy. Arithmetic sequences. khanacademy.org/math/algebra
- Apostol TM. Introduction to Analytic Number Theory. Springer, 1976.
- Weisstein EW. Arithmetic Progression. MathWorld. mathworld.wolfram.com
- Gauss CF. Disquisitiones Arithmeticae. 1801. (Historical origin of the sum formula.)