Arithmetic Sequence Calculator
Calculate the nth term, sum, and first 10 terms of an arithmetic sequence given the first term and common difference.
nth Term (aₙ)
—
Sum (Sₙ)
—
First 10 Terms
What is Arithmetic Sequence Calculator?
An arithmetic sequence (also called an arithmetic progression) is a sequence of numbers where each term after the first is obtained by adding a fixed value called the common difference. This calculator finds the nth term, the sum of the first n terms, and lists the first 10 terms.
How to use
- 1 Enter the first term (a₁) of the sequence.
- 2 Enter the common difference (d) — positive for increasing, negative for decreasing.
- 3 Enter n — the term position you want to calculate.
- 4 The nth term, sum, and first 10 terms appear instantly.
Formula
Example calculation
For a₁ = 5, d = 3, n = 8: a₈ = 5 + 7×3 = 26. S₈ = 8/2 × (10 + 21) = 4 × 31 = 124. First 10 terms: 5, 8, 11, 14, 17, 20, 23, 26, 29, 32.
Frequently asked questions
What is the common difference?
The common difference d is the constant value added to each term to get the next. If d > 0 the sequence increases; if d < 0 it decreases; if d = 0 all terms are equal.
Can d be a fraction or decimal?
Yes. d can be any real number, including fractions and decimals. For example, a₁=1, d=0.5 gives 1, 1.5, 2, 2.5, …
What is the sum of the first n natural numbers?
This is an arithmetic sequence with a₁=1 and d=1. The sum is n(n+1)/2, a special case of the general formula.
How is arithmetic sequence used in real life?
Arithmetic sequences model situations with constant change: loan repayment schedules, salary increments, uniform motion distances, and staircase step heights.
What is the difference between arithmetic and geometric sequences?
Arithmetic sequences add a constant (d) each step. Geometric sequences multiply by a constant (r) each step. 2,4,6,8 is arithmetic (d=2); 2,4,8,16 is geometric (r=2).