Sequence calculator — nth term and sum of arithmetic and geometric sequences

How to use the sequence calculator

1. Under Sequence, pick Arithmetic (add the same amount each time) or Geometric (multiply by the same amount each time).
2. Enter the First term a₁.
3. Enter the Common difference d or the Common ratio r.
4. Enter the Term number n you want.
5. Read nth term aₙ and Sum of n terms Sₙ. Geometric sequences also show Sum to infinity. Check the Formulas box and the list of terms.

What it does and when to use it

Arithmetic and geometric sequences (APs and GPs) are a standard part of school algebra. They also model everyday things: saving a fixed amount each month is arithmetic, while compound growth and repeated halving are geometric. Exam questions ask for a particular term, the sum of the first n terms, or the limit of an infinite series.

For the Fibonacci numbers, use the Fibonacci calculator. To work out how many terms a geometric sequence needs to pass a target, the logarithm calculator solves rⁿ = k. The percentage calculator shows repeated percentage rises, which are geometric with r = 1 + p/100.

How it works

Arithmetic sequence

• aₙ = a₁ + (n − 1)d
• Sₙ = n(a₁ + aₙ) ÷ 2

Geometric sequence

• aₙ = a₁ × rⁿ⁻¹
• Sₙ = a₁(1 − rⁿ) ÷ (1 − r), or n × a₁ when r = 1
• S∞ = a₁ ÷ (1 − r) when |r| < 1

The tool works with exact fractions, so a sequence with d = 0.1 gives 0.1, 0.2, 0.3 exactly rather than 0.30000000000000004.

Step by step for 3, 7, 11, … (a₁ = 3, d = 4) up to n = 10:

1. aₙ = a₁ + (n − 1)d = 3 + (10 − 1) × 4 = 3 + 36 = 39.
2. Sₙ = n(a₁ + aₙ) ÷ 2 = 10 × (3 + 39) ÷ 2 = 10 × 42 ÷ 2 = 210.

The idea behind step 2: pair the first and last terms (3 + 39), the second and second-last (7 + 35), and so on. Each pair is 42, and there are 5 pairs.

Worked examples

Arithmetic. a₁ = 3, d = 4, n = 10: a₁₀ = 39 and Sum of n terms Sₙ: 210.

Doubling. Geometric with a₁ = 3, r = 2, n = 10: nth term aₙ: 1,536, S₁₀ = 3,069, and the sum to infinity diverges.

Halving. a₁ = 1, r = 0.5, n = 10: a₁₀ = 0.001953125, S₁₀ = 1.998046875 and Sum to infinity: 2.

Falling terms. Arithmetic with a₁ = 50, d = −7, n = 8 lists 50, 43, 36, 29, 22, 15, 8, 1, with a sum of 204.

Limits and tips

• n counts from 1, so a₁ is the first term.
• Only the first 12 terms are listed; aₙ and Sₙ are given for any n up to 100,000.
• Very long results are shown in scientific form.
• Everything runs in your browser.