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Arithmetic Sequence Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

An arithmetic sequence calculator is an educational utility that computes individual terms and cumulative sums of arithmetic progressions. It applies the formulas a_n = a_1 + (n-1)d and S_n = (n/2)(2a_1 + (n-1)d) to solve for any unknown parameter, generating a progression table of the first ten terms with step-by-step arithmetic explanations.

Find any specific term (a_n) or the cumulative sum (S_n) of an arithmetic sequence. Features detailed formula substitution steps and an interactive term sequence progression table.

See also: Geometric Sequence Calculator and Summation Calculator.

Quick Answer

Calculate the nth term or sum of an arithmetic sequence. Enter the first term, common difference, and term number to see step-by-step math.

Calculation Target

e.g. 1

e.g. 2

Must be a positive integer. · e.g. 10

Sequence Output

Nth Term (a_10)

19

a₁ = 1 · d = 2

First Term (a₁)1
Common Difference (d)2
Term Number (n)10
Nth Term (a_n)19
Sum of First n Terms (S_n)100

An arithmetic sequence changes by a constant difference (d) at each step.

Step-by-Step Formula Substitution

[1]Identify the parameters of the arithmetic sequence:
[2] - First term (a₁) = 1
[3] - Common difference (d) = 2
[4] - Term number (n) = 10
[5]Apply the arithmetic sequence formula for the nth term:
[6] a_n = a₁ + (n − 1) × d
[7] a_10 = 1 + (10 − 1) × 2
[8] a_10 = 1 + 9 × 2
[9] a_10 = 1 + 18 = 19

First 10 Terms of the Sequence

Term (i)Value (a_i)Cumulative Sum (S_i)
111
234
359
4716
5925
61136
71349
81564
91781
1019100
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Examples

Find the 10th term for first term 1, common difference 2

a_10 = 1 + (10 - 1) × 2 = 19

Find the sum of the first 100 integers (a_1 = 1, d = 1)

S_100 = (100 / 2) × (1 + 100) = 5,050

Arithmetic sequence with negative difference (a_1 = 10, d = -3, n = 5)

a_5 = 10 + (5 - 1) × (-3) = -2 · Sum = 20

How it works

Arithmetic sequences change by adding or subtracting a constant difference:

Formula for the nth Term (a_n)

an = a1 + (n − 1)d

Formula for the Sum of the First n Terms (S_n)

Sn = (n / 2) · (a1 + an)

Sequences vs. Series: What is the Difference?

A sequence is a ordered list of numbers following a specific mathematical pattern. For instance, the sequence of odd numbers is 1, 3, 5, 7, 9.... A series, on the other hand, is the summation of the terms of a sequence. The series corresponding to the odd numbers sequence would be 1 + 3 + 5 + 7 + 9.... This calculator helps you solve for both the individual term points of a sequence and the overall compiled series sum.

Related mathematical calculators

Explore related summation and sequence calculators:

Arithmetic Sequence FormulaRecursive FormulaExplicit FormulaGeometric Sequence CalculatorSummation CalculatorSeries Calculator

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More tools from Education

Geometric Sequence CalculatorCalculate the nth term (a_n), finite sum (S_n), or infinite sum (S) of a geometric sequence. Includes step-by-step derivation and convergence analysis.Summation CalculatorEvaluate sums in sigma notation by specifying the starting index, ending index, and the expression to be summed.

Frequently asked questions

An arithmetic sequence is a list of numbers in which each term after the first is obtained by adding a constant value, called the common difference (d), to the preceding term. For example, 1, 3, 5, 7, 9... is an arithmetic sequence with a common difference of 2.

The common difference is the constant value added to each term in an arithmetic sequence to get the next term. It can be found by subtracting any term from the term that immediately follows it (e.g., d = a_n - a_(n-1)). The common difference can be positive, negative, or zero.

The nth term of an arithmetic sequence is calculated using the formula: a_n = a₁ + (n - 1)d, where a₁ is the first term, d is the common difference, and n is the position of the term in the sequence.

The sum of the first 'n' terms (also called an arithmetic series) can be calculated using the formula: S_n = (n / 2) × (a₁ + a_n), where a₁ is the first term and a_n is the nth term. Alternatively, you can use S_n = (n / 2) × [2a₁ + (n - 1)d].