What this calculator does
Find any term of a number pattern that multiplies by the same amount each step, and add up its terms. That pattern is a geometric sequence. For example, 3, 1.5, 0.75, 0.375, … halves each time, while 1, 2, 4, 8, … doubles. The number you multiply by is called the common ratio, r. Compound interest, radioactive decay, a bouncing ball that keeps a fixed share of its height, and doubling bacteria all follow geometric patterns.
Enter the first term, the ratio and a term number to get that term and the sum of all the terms up to it. When the ratio is between −1 and 1, the calculator also gives the sum of the whole never-ending sequence. When it isn't, it explains why no finite total exists.
Where it's used
Why this math matters outside the classroom:
- Bouncing ball. A ball dropped from 2 m that bounces back to 60% of its height each time rises 2 × 0.6³ = 0.432 m on the third bounce.
- Population growth. A town of 10,000 growing 3% a year has 10,000 × 1.03⁵, about 11,593 people, after 5 years.
- Chessboard grains. Placing 1 grain on the first chessboard square and doubling on each square after totals 2⁶⁴ − 1 grains.
- Equipment value. A $20,000 machine losing 15% of its value each year is worth 20,000 × 0.85³, about $12,282.50, after 3 years.
- Monthly saving. Depositing $100 a month at 0.5% monthly interest builds 100 × (1.005¹² − 1) ÷ 0.005, about $1,233.56, in a year.
Grade level and standards: Grades 9–11; Common Core State Standards for Mathematics HSF.BF.A.2, HSA.SSE.B.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Enter the first term a₁.
- Enter the common ratio r: divide any term by the one before it to find it.
- Enter the term number n you are interested in.
- Read the nth term, the partial sum and the infinite sum, and check the table for the first terms.
What the inputs mean
- First term (a₁)
- The first term. It can be negative or a decimal, but not 0.
- Common ratio (r)
- The factor each term is multiplied by. A ratio between 0 and 1 shrinks the terms; above 1 grows them; a negative ratio makes the signs alternate. It can't be 0.
- Term number (n)
- The position you want, counting the first term as 1, up to 10,000.
Formula and method
aₙ = a₁ × r^(n − 1) Sₙ = a₁ × (1 − rⁿ) / (1 − r), for r ≠ 1 Sₙ = n × a₁, for r = 1 S∞ = a₁ / (1 − r), only when |r| < 1
Here a₁ is the first term, r is the common ratio and n is the position you want. aₙ is the term at that position, Sₙ is the sum of the first n terms, and S∞ is the sum of all the terms forever. |r| means the size of r, ignoring any minus sign.
Reaching position n takes n − 1 multiplications by r, which gives the nth-term formula.
For the partial sum, write Sₙ = a₁ + a₁r + … + a₁rⁿ⁻¹ and multiply the whole line by r. Subtracting the two lines cancels every term except the first and the last, leaving Sₙ(1 − r) = a₁(1 − rⁿ). Dividing by 1 − r gives the formula. When r = 1, every term is a₁, so the sum is n × a₁.
When |r| < 1, rⁿ gets closer and closer to 0 as n grows, so the partial sums approach a₁/(1 − r). When |r| ≥ 1 the terms never shrink toward 0, and the running total either grows without bound or keeps jumping back and forth, so there is no infinite sum.
Worked example
A ball is dropped and its first bounce rises 3 m. Each bounce reaches half the height of the previous one. How high is the 8th bounce, and what is the total height of the first 8 bounces?
- The 8th term: a₈ = 3 × 0.5^7 = 3 / 128 = 0.0234375 m.
- The partial sum: S₈ = 3 × (1 − 0.5^8) / (1 − 0.5) = 6 × (1 − 0.00390625) = 5.9765625 m.
- Because |0.5| < 1, the infinite sum is 3 / (1 − 0.5) = 6 m.
Result: The 8th bounce rises about 0.0234 m, the first 8 bounces total 5.9765625 m, and even infinitely many bounces would never add up to more than 6 m.
Understanding the result
With r > 1 the terms grow exponentially, so even modest term numbers can produce very large values.
With 0 < r < 1 the terms decay toward zero and the partial sums level off near the infinite sum.
A negative ratio makes the terms alternate in sign; the partial sums then zig-zag, settling down only when |r| < 1.
Assumptions and limitations
- Positions start at 1 for the first term.
- The same ratio applies between every pair of neighboring terms.
- Values are computed in floating point, so results with many digits are rounded to about 15 significant figures.
- Terms that become too large to represent produce an error rather than an approximate answer.
- The table shows at most the first 20 terms.
Frequently asked questions
How can an infinite number of terms have a finite sum?
When each term is a fixed fraction of the one before, the pieces shrink fast enough that the total approaches a limit. Adding 1/2 + 1/4 + 1/8 + … gets ever closer to 1 without passing it, which is exactly what a₁/(1 − r) = 0.5/(1 − 0.5) = 1 says.
Why is there no infinite sum when r = −1?
The terms are a₁, −a₁, a₁, −a₁, …, so the running total alternates between a₁ and 0 forever. It never settles on one value, so the series diverges (has no total) even though it does not grow.
How is a geometric sequence related to compound interest?
A balance that grows by a fixed percentage each period is multiplied by the same factor each time, for example 1.05 for 5% growth. The balances form a geometric sequence with that factor as the ratio.
Why can't the first term or the ratio be 0?
With a first term of 0 every term is 0, and with a ratio of 0 every term after the first is 0. Neither gives a meaningful geometric pattern, so both are excluded by the usual definition.
How this calculator is checked
Method: Standard closed forms for the nth term, partial sum and infinite sum of a geometric progression. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
Embed this calculator
Put this Geometric Sequence Calculator on your own website or blog for free. Paste this code where you want it to appear: