Sum of Geometric Sequence Formula Calculator

Calculator Input Form

Calculation mode

First term, a

Common ratio, r

Number of terms, n

Last term, L

Decimal places

Example Data Table

First term	Ratio	Terms	Formula type	Expected result
3	2	5	Finite sum	93
10	0.5	6	Finite sum	19.6875
8	0.25	Infinite	Infinite sum	10.6667
5	-2	4	Nth term	-40

Formula Used

Finite geometric sum: Sₙ = a(1 - rⁿ) / (1 - r), when r is not equal to 1.

Equal ratio case: Sₙ = a × n, when r equals 1.

Nth term: Tₙ = a × rⁿ⁻¹.

Infinite sum: S∞ = a / (1 - r), only when |r| is less than 1.

Estimated ratio: r = (L / a)¹⁄⁽ⁿ⁻¹⁾, where L is the last term.

How to Use This Calculator

Select the calculation mode that matches your problem.
Enter the first term of the geometric sequence.
Enter the common ratio for finite, nth term, or infinite modes.
Enter the number of terms for finite and nth term calculations.
Use the last term field when estimating the ratio.
Select decimal places for the final display.
Press the calculate button to view results above the form.
Download the result as CSV or PDF when needed.

A Geometric Sequence Sum Guide

A geometric sequence grows by multiplying each term by one fixed ratio. The first term starts the pattern. The common ratio controls every step after that. This calculator helps you study that pattern without manual repetition. It handles finite sums, nth terms, estimated ratios, and infinite sums.

Why the Sum Matters

The sum of a geometric sequence appears in many topics. It is used in algebra, finance, physics, computer science, and data growth models. Compound discounts also follow this structure. So do repeated percentage changes. A clear sum helps compare total value, not only the final term.

Finite Sequence Insight

A finite geometric sum adds a set number of terms. The number of terms must be a positive whole number. When the ratio equals one, every term is the same. The sum becomes the first term multiplied by the term count. When the ratio differs from one, the compact geometric formula avoids long addition.

Infinite Sequence Insight

An infinite geometric series only has a usable sum when the ratio is between negative one and one. In that case, terms become smaller over time. The running total approaches a limit. If the ratio is outside that range, the series does not settle into one finite value.

Better Checking With Steps

This tool shows the formula path used for your selected mode. It also reports the nth term, term average, ratio behavior, and a short sequence preview. These details make mistakes easier to see. They also help students explain their work in assignments.

Practical Uses

Use the calculator for loan discount factors, population models, repeated scaling, savings plans, and classroom examples. You can change decimal precision for cleaner results. You can export the result as a CSV file or a PDF file. That makes the calculator useful for records, lessons, and reports.

Careful Input Tips

Use decimal ratios for percentage changes. For example, a ten percent increase uses 1.10. A ten percent decrease uses 0.90. Check negative ratios carefully, because signs alternate between terms. For infinite sums, confirm that the ratio is strictly less than one in absolute value. Small input changes can create very different totals. It supports focused study sessions and faster checking. Use it before final submission.

FAQs

What is a geometric sequence?

A geometric sequence is a list of numbers where each term is made by multiplying the previous term by a fixed common ratio.

What does the first term mean?

The first term is the starting value of the sequence. It is usually written as a in the geometric sum formula.

What is the common ratio?

The common ratio is the multiplier between consecutive terms. Divide any term by the previous term to find it.

When should I use the finite sum formula?

Use the finite sum formula when the sequence has a fixed number of terms. Enter the first term, ratio, and term count.

When does an infinite geometric sum exist?

An infinite geometric sum exists only when the absolute value of the ratio is less than one. Then the terms shrink toward zero.

What happens when the ratio equals one?

Every term becomes the same value. The sum is simply the first term multiplied by the number of terms.

Can this calculator handle negative ratios?

Yes. Negative ratios create alternating signs. The calculator shows the behavior and previews the first terms for easier checking.

Why export results as CSV or PDF?

CSV works well for spreadsheets. PDF works well for printing, sharing, or saving clean calculation records for lessons and reports.