geometric sum calculator

What is a geometric sum?

A geometric sum is what you get when you add terms in a geometric sequence. In a geometric sequence, each term is multiplied by the same constant ratio to get the next one. That constant is called the common ratio, usually written as r.

For example, if the first term is 3 and the ratio is 2, the terms are:

• 3
• 6
• 12
• 24
• ...

The sum of the first few terms is called a finite geometric sum. If the pattern continues forever and still settles toward a limit, that is an infinite geometric series.

Core formulas used by this calculator

Finite geometric sum

Given first term a, ratio r, and number of terms n:

• If r ≠ 1: S_n = a(1 - rⁿ) / (1 - r)
• If r = 1: S_n = a × n

This closed-form formula is extremely useful because it avoids adding term-by-term manually.

Infinite geometric sum

The infinite series a + ar + ar² + ... converges only when |r| < 1. In that case:

• S_∞ = a / (1 - r)

If |r| ≥ 1, the infinite sum does not converge to a finite value.

How to use this geometric sum calculator

• Enter your first term a.
• Enter the common ratio r.
• Enter the number of terms n as a positive integer.
• Click Calculate Sum.
• Read the finite sum, nth term, preview terms, and convergence status.

Worked examples

Example 1: Growth pattern

Suppose you model a sequence with a = 2, r = 3, and n = 5. The terms are 2, 6, 18, 54, 162. The total is 242.

Using the formula: S₅ = 2(1 - 3⁵)/(1 - 3) = 242.

Example 2: Decay pattern

Let a = 100, r = 0.8, n = 6. This could model shrinking values over time. The calculator quickly gives both the exact finite sum and the infinite limit because |0.8| < 1.

Where geometric sums are used in real life

• Finance: discounted cash flow and annuity-style calculations.
• Investing: repeated percentage growth or decline models.
• Physics: diminishing bounces, attenuation, and wave reflections.
• Computer science: divide-and-conquer runtime analysis and recursive costs.
• Signal processing: filter behavior and repeated scaling effects.

Common mistakes to avoid

• Using a non-integer value for n in a finite sum.
• Forgetting the special case when r = 1.
• Applying the infinite-series formula when |r| ≥ 1.
• Mixing up arithmetic and geometric sequences.

Quick FAQ

Can the ratio be negative?

Yes. A negative ratio creates alternating signs in the terms, and the calculator handles that correctly.

What happens if r is very close to 1?

The sum can grow nearly linearly with n in that case. The calculator automatically switches behavior for the exact r = 1 case.

Why show only a preview of terms?

For large n, listing every term is not practical. The calculator shows the first several terms so you can verify the pattern instantly.