Geometric Sequence Calculator: nth Term and Sum
List the terms of a geometric sequence from its first term and common ratio, with the nth-term formula and the exact sum. Ratios such as 1/3 or -0.5 stay exact. Nothing is uploaded.
- Formula for aₙ
- aₙ = a₁ · r^(n - 1) = 1 · 2^(n - 1)
- Term a₁₀
- 512
- Formula for Sₙ
- Sₙ = a₁(1 - rⁿ) / (1 - r)
- Sum S₁₀
- 1023
About the geometric sequence calculator
This page lists the terms of a geometric sequence and works out its nth term and sum. Enter the first term a₁, the common ratio r and the number of terms n, and the terms appear at once, together with the formula for the nth term, the value of the last term aₙ and the sum Sₙ. The ratio may be a whole number, a decimal such as -0.5 or a fraction such as 2/3, and every term is exact: terminating values print as decimals and the rest as fractions, so a ratio of 1/3 gives 1, 1/3, 1/9, 1/27 instead of rounded approximations. Everything is calculated in your browser, and nothing is uploaded.
How to use it
Steps
- Type the First term a₁, for example 3.
- Type the Common ratio r, for example 2. A ratio between -1 and 1 makes the terms shrink towards zero; a negative ratio makes them alternate in sign.
- Set the Number of terms, up to 10,000. Very fast-growing sequences stop earlier, at 1,000,000 characters of output.
- Read the formula, the last term and the sum, then copy or download the list with a comma, space or new-line separator.
The formulas
nth term
Each term is the previous one times r, so the nth term is
With a₁ = 3 and r = 2 the page writes aₙ = 3 · 2^(n - 1), so a₁₀ = 1536.
Sum of the first n terms
Multiplying the sum by r shifts every term along by one place, and subtracting the two sums leaves only the first and last terms. For r ≠ 1 that gives
and for r = 1 every term equals a₁, so Sₙ = n · a₁. For 3, 6, 12, 24, 48 the sum is 3 × (1 − 32) / (1 − 2) = 93. When |r| < 1 the sums approach a₁ / (1 − r) as n grows, the value of the infinite geometric series.
Worked examples
- Doubling: one grain of rice on the first square of a chessboard and twice as many on each next square gives 2⁶⁴ − 1 = 18446744073709551615 grains in all, shown here digit for digit.
- Halving: a first term of 8 and a ratio of -0.5 give 8, -4, 2, -1, 0.5.
- Thirds: a first term of 1 and a ratio of 1/3 give a sum of 40/27 after four terms, edging towards 3/2.
Privacy
The terms and sums are worked out by code running on this page, so the values you type are never sent anywhere and are forgotten when you close the tab.
Related tools
For a sequence that adds a fixed difference instead, use the Arithmetic Sequence Calculator. Powers of two and ten with negative exponents, and many other sequences, are in the full Number Sequence Generator, and the Exponent Calculator raises a single number to a power.
Open-source credits
The arithmetic is implemented in-house in TypeScript with the built-in BigInt type and exact fractions in lowest terms; no third-party library is used. The powers of two and ten built on the same code are tested against OEIS A000079 and A011557.