Geometric progression calculator
The n-th term, the sum of the series and the terms themselves.
Fill in the fields and the result will appear here automatically.
Find the nth term and the sum of the first n terms when each term is the previous one multiplied by r. This page accepts finite a₁, nonzero finite r and an integer n from 1 to 50. a₁ may be negative or zero; negative r alternates the signs of nonzero terms. The table previews twenty terms. For |r| < 1 an infinite-series sum is also shown. Intermediate binary terms and their finite sum are retained exactly until final rounding.
How it works
Formula and logic
aₙ = a₁rⁿ⁻¹. For r ≠ 1, Sₙ = a₁(1−rⁿ)/(1−r); for r = 1, Sₙ = na₁. This implementation sums exact intermediate binary terms, avoiding subtraction of nearly equal powers when r is close to 1. For |r| < 1, S∞ = a₁/(1−r).
Example
Starting at 2 with a ratio of 3, the tenth term is 39,366 and the series sums to 59,048.
Fields and units
- First term — unitless
- Common ratio — unitless
- Number of terms — unitless
How to use
- — Enter the first term — it may be negative.
- — Enter the ratio: 2 doubles each step, 0.5 halves it.
- — Enter how many terms you need, up to fifty.
- — The table lists the first twenty terms.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: constant ratios and the nth-term geometric-sequence formula OpenStax: finite progression sums and the infinite geometric sum for |r| < 1
- Limitation
- The 10¹⁵ limit applies to the nth term and finite sum, not the additional infinite sum. Every displayed value must be finite without a nonzero value rounding to zero. Decimal inputs and displayed results are rounded.
FAQ
Why is a ratio of zero rejected?
This is a page restriction. The recurrence a₁, 0, 0, … with r = 0 is mathematically meaningful, but this tool retains its nonzero-ratio input rule.
When does the infinite sum exist?
For a nonzero first term, the series converges when |r| < 1, with S∞ = a₁/(1−r). This page shows that row only under this condition. If a₁ = 0 the sequence is zero even at other ratios, but no separate infinite-sum row is added.
Can the ratio be negative?
Yes. With nonzero a₁ the signs alternate, and the same sum formulas apply. If a₁ = 0 every term remains zero.
Why fifty terms and not more?
The range 1–50 is a page limit, not a limit of the mathematical formula. The page also requires |aₙ| and |Sₙ| to be below 10¹⁵. These limits bound computation and output.
Is a progression the same thing as compound interest?
With a constant rate per period, r = 1 + rate and an amount with no additional payments grows geometrically. Periods must match; financial calculators separately model contributions, compounding frequency and monetary rounding.