Sequence Calculator
A sequence is an ordered list of numbers that follows a rule. In an arithmetic sequence you add the same amount each step (3, 7, 11, 15…). In a geometric sequence you multiply by the same amount each step (2, 6, 18, 54…). This calculator finds any term, the sum of the first n terms and, for geometric sequences with a small ratio, the sum to infinity.
How to use the sequence calculator
- Choose arithmetic or geometric.
- Enter the first term and the common difference or ratio, then the term number n.
- Read the n-th term, the sum of the first n terms and the first ten terms.
Formula
- Arithmetic: aₙ = a₁ + (n − 1)d and Sₙ = n(2a₁ + (n − 1)d) ÷ 2
- Geometric: aₙ = a₁ × rⁿ⁻¹ and Sₙ = a₁(1 − rⁿ) ÷ (1 − r)
- Infinite geometric sum (|r| < 1): S = a₁ ÷ (1 − r)
Worked examples
Arithmetic
Starting at 3 and adding 4: the 10th term is 3 + 9 × 4 = 39 and the sum of ten terms is 10 × (6 + 36) ÷ 2 = 210.
Geometric
Starting at 2 and multiplying by 3: the 6th term is 2 × 3⁵ = 486 and the sum is 728.
Infinite series
1 + ½ + ¼ + … adds up to 1 ÷ (1 − 0.5) = 2.
Sequences in the real world
Arithmetic sequences
Anything that grows by a fixed amount: saving $50 more each month, a taxi fare with a fixed per-mile rate, seats in the rows of a theatre that grow by two each row. The sum of an arithmetic sequence is the average of the first and last term times the number of terms, which is the trick young Gauss used to add 1 to 100 (5,050).
Geometric sequences
Anything that grows by a fixed percentage: compound interest, population growth, a bouncing ball that loses part of its height each time, or the doubling of a viral post.
| Situation | Type | Rule |
|---|---|---|
| Salary rises $2,000 a year | Arithmetic | d = 2,000 |
| Investment grows 5% a year | Geometric | r = 1.05 |
| Ball rebounds to 80% of its last height | Geometric | r = 0.8 |
Converging series
Zeno’s paradox adds ½ + ¼ + ⅛ + … and the sum is exactly 1. Infinite geometric series with |r| < 1 have a finite sum a₁ ÷ (1 − r), which is used in physics, finance and computing. Compare with how compound interest works.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- An arithmetic sequence starts at 5 with difference 3. Find the 20th term and the sum of 20 terms.
Show answer
Term 20 (aₙ): 62 · Sum of the first 20 terms: 670
- A geometric sequence starts at 4 with ratio 2. Find the 8th term and the sum.
Show answer
Term 8 (aₙ): 512 · Sum of the first 8 terms: 1,020
Common mistakes to avoid
- Using n instead of n − 1 in the exponent or the multiplier of d.
- Trying to sum a geometric sequence to infinity when |r| ≥ 1, which diverges.
- Mixing up the two types: check whether you add or multiply.
Frequently asked questions
How do I tell arithmetic from geometric?
Compare consecutive terms. A constant difference means arithmetic; a constant ratio means geometric.
What is a series?
The sum of the terms of a sequence.
When does an infinite geometric series converge?
When the absolute value of the ratio is less than 1.
Where are sequences used?
In saving plans, loan schedules, population growth and patterns in nature.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.