acalculator

What is my geometric sequence?

Type the first term, the common ratio, and n. The geometric sequence calculator finds the nth term, the sum of the first n terms, and the sum to infinity when it exists, with the first terms and the formulas.

Your numbers

nth term (aₙ)
1,024

Term 10 is 1,024, and the sum of terms 1 to 10 is 2,046.

Sum of the first n terms (Sₙ)
2,046
First terms
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Explicit formula
a_n = 2 × 2^(n - 1)
Recursive formula
a_1 = 2, a_n = 2 × a_(n - 1)

nth term (aₙ): 1,024. Term 10 is 1,024, and the sum of terms 1 to 10 is 2,046.

How to calculate

Finds the nth term, the sum of the first n terms, and the sum to infinity of a geometric sequence from its first term and common ratio, with the first terms and the formulas.

Example with the default inputs (First term (a₁) 2, Common ratio (r) 2, Term number (n) 10): Term 10 is 1,024, and the sum of terms 1 to 10 is 2,046.

Method: aₙ = a₁ × r^(n − 1); Sₙ = a₁(1 − rⁿ)/(1 − r), or n × a₁ when r = 1; for −1 < r < 1 the sum to infinity is a₁/(1 − r).

  • Terms are numbered from 1; the first term is a₁.
  • The numbers are read as the exact decimals typed and the formulas are worked out in exact fractions; each result is then the nearest 64-bit float.
  • A term or sum beyond about 1.8 × 10^308 has no answer.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. First term (a₁) 8, Common ratio (r) -0.5, Term number (n) 11 gives nth term (aₙ) 0.007813, Sum of the first n terms (Sₙ) 5.335938, Sum to infinity 5.333333, Recursive formula a_1 = 8, a_n = (-0.5) × a_(n - 1).Source: OpenStax, Algebra and Trigonometry 2e, section 13.4 Series and Their Notations (Sₙ = a₁(1 − rⁿ)/(1 − r), S = a₁/(1 − r) for |r| < 1), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations, retrieved 2026-10-02, example: 8 + (−4) + 2 + … has S₁₁ ≈ 5.336
  2. First term (a₁) 248.6, Common ratio (r) 0.4, Term number (n) 3 gives nth term (aₙ) 39.776, Sum of the first n terms (Sₙ) 387.816, Sum to infinity 414.333333, First terms 248.6, 99.44, 39.776.Source: OpenStax, Algebra and Trigonometry 2e, section 13.4 Series and Their Notations (Sₙ = a₁(1 − rⁿ)/(1 − r), S = a₁/(1 − r) for |r| < 1), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations, retrieved 2026-10-02, example: 248.6 + 99.44 + 39.776 + … = 414.3̅
  3. First term (a₁) 3, Common ratio (r) 2, Term number (n) 8 gives nth term (aₙ) 384, Sum of the first n terms (Sₙ) 765, First terms 3, 6, 12, 24, 48, 96, 192, 384, Explicit formula a_n = 3 × 2^(n - 1).Source: OpenStax, Algebra and Trigonometry 2e, section 13.3 Geometric Sequences (aₙ = a₁rⁿ⁻¹), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-3-geometric-sequences, retrieved 2026-10-02
  4. First term (a₁) 5, Common ratio (r) -3, Term number (n) 4 gives nth term (aₙ) -135, Sum of the first n terms (Sₙ) -100.

How it works

This page is the geometric setting of the sequence calculator, with the same inputs, rules and outputs. Terms are numbered from 1. With first term a₁, common ratio r, and term number n:

  • nth term: aₙ = a₁ × r^(n − 1), with r^0 = 1 (also when r is 0)
  • sum of the first n terms: Sₙ = a₁(1 − rⁿ)/(1 − r) when r is not 1, and Sₙ = n × a₁ when r = 1
  • sum to infinity, only when −1 < r < 1: S∞ = a₁/(1 − r); otherwise it is not shown

The numbers are read as the exact decimals typed (0.4 is exactly two fifths), the formulas are worked out in exact fractions, and each result is then the nearest 64-bit floating-point number, shown to at most 6 decimal places. A term or sum beyond the largest 64-bit float (about 1.8 × 10^308) has no answer: "The nth term is too large to show." or "The sum is too large to show." The first term is from −10^9 to 10^9, the ratio from −1,000 to 1,000, and n a whole number from 1 to 10,000.

The calculator also shows:

  • First terms: a₁, a₂, … up to 10 terms (n terms when n is less than 10). Each is written as a decimal in full (no exponent, no thousands separators, a plain hyphen for a minus sign) when it needs at most 20 decimal places and at most 30 digits, and otherwise as the shortest text of its nearest 64-bit float in JavaScript’s number notation (1e+25). They are joined by a comma and a space, followed by , … when n is more than 10.
  • Explicit formula: a_n = a₁ × r^(n - 1), with a₁ and r written as the terms are, a negative r in brackets ((-3)^(n - 1)), and a₁ × left out when a₁ is 1.
  • Recursive formula: a_1 = a₁, a_n = r × a_(n - 1), a negative r in brackets.

Worked examples by hand

a₁ = 8, r = −0.5, n = 11. a₁₁ = 8 × (−1/2)¹⁰ = 8/1024 = 0.0078125. S₁₁ = 8 × (1 − (−1/2)¹¹)/(1 + 1/2) = 8 × (2049/2048) × (2/3) = 5.3359375. Since −1 < −0.5 < 1, S∞ = 8 ÷ 1.5 = 5.333….

a₁ = 248.6, r = 0.4, n = 3. 248.6, 99.44, 39.776; S₃ = 387.816; S∞ = 248.6 ÷ 0.6 = 414.333….

a₁ = 3, r = 2, n = 8. 3, 6, 12, 24, 48, 96, 192, 384; S₈ = 3 × (2⁸ − 1)/(2 − 1) = 765; explicit formula a_n = 3 × 2^(n - 1).

a₁ = 5, r = −3, n = 4. 5, −15, 45, −135; S₄ = 5 × (1 − 81)/(1 + 3) = −100.

Other questions people ask

What is a geometric sequence?

A list of numbers where each term is the one before times the same number, the common ratio r. 3, 6, 12, 24 has r = 2; 8, −4, 2, −1 has r = −1/2, so its signs alternate.

How do I find the nth term?

Use aₙ = a₁ × r^(n − 1). For 3, 6, 12, …, the 8th term is 3 × 2⁷ = 384.

How do I find the common ratio?

Divide any term by the one before it: 6 ÷ 3 = 2. Check a second pair to be sure the ratio stays the same; if it does not, the sequence is not geometric.

How do I add up the first n terms?

Use Sₙ = a₁(1 − rⁿ)/(1 − r) when r is not 1. For 8 + (−4) + 2 + … (11 terms), S₁₁ = 8(1 − (−1/2)¹¹)/(1 + 1/2) = 5.3359375. When r = 1 every term is a₁, so Sₙ = n × a₁.

When does a geometric series have a sum to infinity?

Only when −1 < r < 1. The terms then shrink toward 0, and the sums approach a₁/(1 − r). For 248.6 + 99.44 + 39.776 + … (r = 0.4), the sum to infinity is 248.6 ÷ 0.6 = 414.33…. With r = 2 or r = −1 the sums never settle, so there is none.

Why does the sum to infinity matter?

It turns an endless repeating pattern into one number. A repeating decimal is an example: 0.333… is 0.3 + 0.03 + 0.003 + …, a geometric series with r = 0.1, whose sum is 0.3 ÷ 0.9 = 1/3.