Arithmetic Sequence Calculator
Calculate the nth term and sum of an arithmetic sequence from its first term, common difference, and a positive whole-number term count.
- Sum of n Terms
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Formula: aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ) ÷ 2
Use a positive whole number for n. The common difference may be positive, negative or zero.
Use the Arithmetic Sequence Calculator to find a specific term and the sum of the first n terms. Enter the first term, the common difference, and a positive whole-number term position. The calculation is deterministic: the same three inputs always produce the same two results.
The first term and common difference can be positive, zero, negative, or decimal values. The term number n must be a positive whole number because it identifies a position in the sequence. A negative common difference creates a decreasing arithmetic sequence and is valid.
How to Use the Arithmetic Sequence Calculator
- Enter the first term, written as a1.
- Enter the common difference, written as d. This is the fixed amount added to move from one term to the next.
- Enter a positive whole number for the term position n.
- Select Calculate to see the nth term and the sum of the first n terms.
Keep all values in compatible units. If the terms represent dollars, for example, both the first term and common difference should be dollar amounts.
Arithmetic Sequence Formulas and Worked Examples
a_n = a_1 + (n - 1)d
S_n = n/2 × [2a_1 + (n - 1)d]The explicit formula a_n = a_1 + (n - 1)d finds the term at position n. The recursive description is a_1 = first term and a_n = a_(n-1) + d for later terms. The sum formula adds the finite series from the first term through the nth term. OpenStax’s arithmetic-sequences section covers the common difference plus recursive and explicit forms, while its series section gives the finite arithmetic-series sum formula.
For a1 = 3, d = 2, and n = 5, the sequence begins 3, 5, 7, 9, 11. The fifth term is 3 + (5 - 1) × 2 = 11, and the sum is 5/2 × [2 × 3 + (5 - 1) × 2] = 35.
A decreasing example also works. With a1 = 20, d = -3, and n = 4, the terms are 20, 17, 14, 11. The fourth term is 11 and their sum is 62.
Valid Inputs and Related Calculators
The calculator rejects zero, negative, and fractional values of n. Very large values or long decimal inputs can be rounded by normal browser number arithmetic, so treat the displayed decimals as calculated approximations when precision matters.
Arithmetic sequences change by a constant amount. If values instead change by a constant ratio, use the Geometric Sequence Calculator. Use the Sum Calculator to add a separate list of values, or the Slope-Intercept Calculator when you need the equation of a straight line.
Common uses include checking algebra exercises, modelling a fixed increase or decrease per period, and verifying totals for evenly spaced values. The result is a mathematical calculation, not a prediction about future growth.
Frequently asked questions
What is an arithmetic sequence?
An arithmetic sequence is a sequence in which the difference between each pair of consecutive terms is constant. That fixed value is the common difference.
How do I find the nth term of an arithmetic sequence?
Use a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is a positive whole-number position.
What does the sum of n terms mean?
It is the total of the finite series from the first term through the nth term. This calculator uses S_n = n/2 × [2a_1 + (n - 1)d].
Can this tool handle negative common differences?
Yes. A negative common difference produces a decreasing arithmetic sequence. The first term and common difference may also be zero or decimal values.
Is there a limit to the number of terms I can calculate?
The term count must be a positive whole number. Extremely large inputs can lose precision because the calculator uses standard browser number arithmetic.
What happens if I enter an invalid term number?
The calculator asks you to correct n when it is zero, negative, fractional, or not a finite number. No sequence result is calculated until n is a positive whole number.
Can I use this calculator for geometric sequences?
No. A geometric sequence changes by a constant ratio rather than a constant difference. Use the Geometric Sequence Calculator for that pattern.