Geometric Sum Calculator

Q: What is a geometric series?

A geometric series is a sum of terms where each term is multiplied by a constant ratio (r) to get the next term. Example: 2, 6, 18, 54 has ratio r = 3.

Q: When does a geometric series converge?

An infinite geometric series converges (has a finite sum) only when |r| < 1. The infinite sum is S∞ = a / (1 − r). If |r| ≥ 1, the series diverges.

Q: What if the common ratio is 1?

When r = 1, every term equals a, so the sum is simply Sₙ = a × n.

Q: How do I add this Geometric Sum Calculator to my site?

Absolutely. Use the "Embed" option above to tailor the dimensions, color scheme, and styling to match your site. Copy the generated iframe snippet and drop it into your HTML, WordPress editor, or any CMS. There is no cost and no account required. See calculory.com/services/embed-calculators for a step-by-step walkthrough.

Calculate the sum of a geometric series. Enter the first term, common ratio, and number of terms to get the finite sum, infinite sum (if convergent), and term list.

Enter Values

First Term (a)

Common Ratio (r)

Number of Terms (n)

Result

Enter values above and click Calculate to see your result.

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Formula

Core Formula

S_n = \frac{a(1 - r^n)}{1 - r}

How it works: For a geometric series with first term a, common ratio r, and n terms: multiply a by (1 − rⁿ), then divide by (1 − r). If |r| < 1, the infinite sum converges to a / (1 − r).

Worked Example

First term: 3, Ratio: 2, Terms: 5

Series: 3, 6, 12, 24, 48

1Step 1: S₅ = 3(1 − 2⁵) / (1 − 2)

2Step 2: S₅ = 3(1 − 32) / (−1)

3Step 3: S₅ = 3 × 31 = 93

Understanding Geometric Series Sums

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The sum of such a series, whether finite or infinite, has wide-ranging applications in mathematics, finance, and various scientific fields. Calculating the finite sum involves using a specific formula that accounts for the first term, the common ratio, and the total number of terms. This sum represents the total value accumulated after a certain number of steps in the series.

For an infinite geometric series, the sum does not always exist. It only converges to a finite value if the absolute value of its common ratio is less than one. When this condition, |r| < 1, is met, the terms of the series gradually become smaller and smaller, approaching zero, allowing the entire infinite series to sum up to a specific number. If |r| is greater than or equal to one, the terms either grow larger or stay the same size, causing the sum to diverge to infinity, meaning it has no finite sum. Understanding these concepts is crucial for solving problems related to compound interest, depreciation, and even the decay of radioactive materials.

Geometric sums are used in finance for calculating compound interest and loan repayments.
They model physical phenomena like pendulum swings and radioactive decay.
The sum of an infinite geometric series only exists if the common ratio's absolute value is less than one.
Each term in a geometric series is found by multiplying the previous term by a constant common ratio.

Grasping geometric sums helps in analyzing various growth and decay patterns. Use the calculator to explore how different inputs affect both finite and infinite series sums.

You can also calculate changes using our Taylor Polynomial Calculator.

Frequently Asked Questions

What is a geometric series?

A geometric series is a sum of terms where each term is multiplied by a constant ratio (r) to get the next term. Example: 2, 6, 18, 54 has ratio r = 3.

When does a geometric series converge?

An infinite geometric series converges (has a finite sum) only when |r| < 1. The infinite sum is S∞ = a / (1 − r). If |r| ≥ 1, the series diverges.

What if the common ratio is 1?

When r = 1, every term equals a, so the sum is simply Sₙ = a × n.

How do I add this Geometric Sum Calculator to my site?

Absolutely. Use the "Embed" option above to tailor the dimensions, color scheme, and styling to match your site. Copy the generated iframe snippet and drop it into your HTML, WordPress editor, or any CMS. There is no cost and no account required. See calculory.com/services/embed-calculators for a step-by-step walkthrough.

AI Assistant

Ask about this calculator

I can help you understand the geometric sum calculator formula, interpret your results, and answer follow-up questions.

Try asking

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