Sequences Calculator

by
DC
Expert Reviewer: David Chen, CFA
Last updated: December 2025

Welcome to the **Geometric Sequence Sum Calculator**—a precise tool for solving complex sequence problems. Enter three of the four variables ($a_1, r, n, S_n$) to instantly find the missing value. The logic accounts for consistency and special boundary conditions, ensuring accurate results for all geometric progressions.

Geometric Sequence Sum Calculator

Geometric Sequence Sum Formula

$$S_n = a_1 \frac{1 – r^n}{1 – r}, \quad r \ne 1$$

Formula Source: Wolfram MathWorld | Wikipedia

Variables

The calculator uses four core variables related to a geometric sequence:

  • First Term ($a_1$): The initial number in the sequence.
  • Common Ratio ($r$): The constant factor between consecutive terms.
  • Number of Terms ($n$): The total count of terms being summed.
  • Sum of $n$ Terms ($S_n$): The total accumulation of all terms in the sequence.

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What is a Geometric Sequence Sum Calculator?

A Geometric Sequence Sum Calculator is an indispensable tool used in finance, mathematics, and engineering to quickly determine the total value of a geometric progression. A geometric sequence is a list 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 ($r$).

This calculator automates the process of solving for any unknown variable when three of the four core values—the first term ($a_1$), the common ratio ($r$), the number of terms ($n$), or the total sum ($S_n$)—are known. It is crucial for modeling scenarios like compound interest, exponential growth, and decay.

The tool is designed to handle special cases, such as when the common ratio ($r$) equals 1, and also performs consistency checks if all four variables are entered, ensuring the input set is mathematically valid.

How to Calculate the Geometric Sum (Example)

Suppose you are calculating the sum of 5 terms where the first term is 10 and the common ratio is 2. The sequence is 10, 20, 40, 80, 160.

  1. Identify Variables: $a_1 = 10$, $r = 2$, $n = 5$. $S_n$ is unknown.
  2. Apply Formula: $$S_n = 10 \cdot \frac{1 – 2^5}{1 – 2}$$
  3. Calculate Exponent: $2^5 = 32$.
  4. Substitute and Simplify: $$S_n = 10 \cdot \frac{1 – 32}{1 – 2} = 10 \cdot \frac{-31}{-1}$$
  5. Final Result: $$S_n = 10 \cdot 31 = 310$$. The sum is 310.

Frequently Asked Questions (FAQ)

What is the difference between an arithmetic and a geometric sequence?
Can I calculate the sum if the common ratio ($r$) is negative?
What happens if the common ratio ($r$) is exactly 1?
Why do I need to input at least three variables?
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