Quick answer
In an arithmetic sequence aₙ = a₁ + (n − 1)d and the sum is Sₙ = n(a₁ + aₙ) ÷ 2. With a₁ = 2 and d = 3, the 10th term is 29 and the sum of ten terms is 155.
Enter the first term, common difference and term number. The calculator returns the nth term and the sum.
Related tools: Number Sequence Calculator: Identify an arithmetic or geometric sequence and find the next terms. Geometric Sequence Calculator: nth term and sum of a geometric sequence. Slope Calculator: Slope, angle, distance and line equation through two points.
How to use the Arithmetic Sequence Calculator
- Enter first term (a₁).
- Enter common difference (d).
- Enter term number (n) (a whole number, between 1 and 1000000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| First term (a₁) | A number | — |
| Common difference (d) | A number | — |
| Term number (n) | A whole number | ≥ 1, ≤ 1000000 |
Formula and method
Formula
aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ) ÷ 2
IEEE-754 double precision; shown to at most 6 decimal places.
Closed-form formulas.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: First term (a₁) = 2; Common difference (d) = 3; Term number (n) = 10.
| Result | Value |
|---|---|
| a10 | 29 |
| Sum of n terms | 155 |
Example 2
Inputs: First term (a₁) = 5; Common difference (d) = -2; Term number (n) = 4.
| Result | Value |
|---|---|
| a4 | -1 |
| Sum of n terms | 8 |
The first line is the headline result (a10). Below it the arithmetic Sequence Calculator lists sum of n terms, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
Terms: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. a₁₀ = 2 + 9 × 3 = 29. Sum = 10 × (2 + 29) ÷ 2 = 155. Gauss's trick of pairing first and last terms gives the same.
Applications
Arithmetic sequences model steady change: savings with fixed deposits, stacked seating rows and linear depreciation. n must be at least 1. The difference can be negative for decreasing sequences.
How the result changes with first term
Holding every other input at the first worked example, this table changes only first term. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| First term | a10 | Sum of n terms |
|---|---|---|
| 1 | 28 | 145 |
| 2 | 29 | 155 |
| 4 | 31 | 175 |
| 8 | 35 | 215 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Term number (n): entering "0" shows "Term number (n) must be at least 1." beside that field and no result.
Choosing between the Arithmetic Sequence Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Number Sequence Calculator](/math/number-sequence-calculator/) | sequence | Pattern |
| [Geometric Sequence Calculator](/math/geometric-sequence-calculator/) | first term, common ratio and term number | a5 |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- n ≥ 1
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for term number (n) to confirm this.
Frequently asked questions
What is the common difference?
The amount added each step.
How do I find n from a term?
n = (aₙ − a₁) ÷ d + 1.
Is the sequence linear?
Yes, terms lie on a line.
What does the Arithmetic Sequence Calculator calculate?
nth term and sum of an arithmetic sequence. It returns a10 and Sum of n terms.
What formula does the Arithmetic Sequence Calculator use?
aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ) ÷ 2. Closed-form formulas.
What do I need to enter in the arithmetic Sequence Calculator?
You enter First term (a₁), Common difference (d) and Term number (n).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. n ≥ 1.
What is an example of a arithmetic Sequence result?
With first term (a₁) 2, common difference (d) 3 and term number (n) 10, the a10 is 29.
Related calculators
- Number Sequence Calculator — Identify an arithmetic or geometric sequence and find the next terms.
- Geometric Sequence Calculator — nth term and sum of a geometric sequence.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Prime Factorization Calculator — Break a whole number into its prime factors.
- Exponent Calculator — Raise a base to any power.
- Perfect Square Calculator — Check whether a number is a perfect square.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.