Math

Geometric Sequence Calculator (aₙ = a₁·rⁿ⁻¹)

Enter the first term a₁ and common ratio r to compute the nth term, partial sums, and infinite sum (when |r| < 1). Includes a term table and bar chart.

Tips

  • When r > 1 the sequence grows rapidly. When 0 < r < 1 it approaches 0. When r < 0 the terms alternate in sign (oscillating sequence).
  • When |r| < 1, the infinite sum converges to S∞ = a₁ / (1 − r). For example with a₁ = 1 and r = 1/2, S∞ = 2.
  • The nth term is aₙ = a₁ · r^(n−1) and the partial sum is Sₙ = a₁(1 − rⁿ) / (1 − r) (when r ≠ 1).
  • The bar chart shows each term as a bar. For large r the bars grow very quickly, so r values between 1.1 and 1.5 make the growth pattern easiest to see.

FAQ

The infinite sum S∞ = a₁ / (1 − r) is finite only when |r| < 1. When |r| ≥ 1 the terms do not shrink fast enough and the sum diverges (goes to infinity or oscillates without bound).

A negative common ratio produces an oscillating sequence where terms alternate in sign. For example, a₁ = 1 and r = −2 gives 1, −2, 4, −8, 16, …

Let S = a₁ + a₁r + … + a₁r^(n−1). Multiply both sides by r: rS = a₁r + … + a₁rⁿ. Subtract: S − rS = a₁ − a₁rⁿ, giving S(1−r) = a₁(1−rⁿ), so Sₙ = a₁(1−rⁿ)/(1−r).
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Side Note — Geometric growth and the paper-folding myth

A classic illustration of geometric growth: if you fold a 0.1 mm sheet of paper in half repeatedly, after n folds its thickness is 0.1 × 2ⁿ mm. The Moon is about 384,400 km away — roughly 3.844 × 10¹¹ mm — so 42 folds would theoretically reach the Moon (2⁴² ≈ 4.4 × 10¹²).

In practice, paper stiffness limits a standard sheet to about 7–8 folds. In 2012, high-school student Britney Gallivan set a world record by folding a long strip of paper 12 times. The mathematical reality remains: geometric sequences grow far beyond our intuition, which is why compound interest, viral spread, and population growth all follow the same pattern.

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