Geometric Sequence Calculator

Calculate nth term, sum of terms, and infinite series

Enter the first term, common ratio, and number of terms to calculate geometric sequence properties including the nth term and sum of the sequence.

The initial value of the sequence

Each term is multiplied by this ratio to get the next term

Must be a positive integer

Example Problems

Try these common geometric sequence problems

Basic Geometric Sequence

basic

Simple sequence with ratio 2

First Term (a₁): 2

Common Ratio (r): 2

Number of Terms (n): 5

Calculation Type: All Calculations

Decimal Common Ratio

decimal

Sequence with fractional ratio

First Term (a₁): 100

Common Ratio (r): 0.5

Number of Terms (n): 6

Calculation Type: All Calculations

Negative Common Ratio

negative

Alternating sequence

First Term (a₁): 3

Common Ratio (r): -2

Number of Terms (n): 4

Calculation Type: All Calculations

Infinite Series

infinite

Sum of infinite geometric series

First Term (a₁): 1

Common Ratio (r): 0.25

Number of Terms (n): 10

Calculation Type: Infinite Series Sum

Other Titles
Understanding Geometric Sequences: A Comprehensive Guide
Master the fundamentals of geometric sequences, series, and their applications in mathematics

What is a Geometric Sequence?

  • Definition and Properties
  • Common Ratio Concept
  • Sequence vs Series
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This mathematical concept is fundamental in algebra, calculus, and many real-world applications.
Key Properties of Geometric Sequences
The defining characteristic of a geometric sequence is the constant ratio between consecutive terms. If we have a sequence a₁, a₂, a₃, ..., then r = a₂/a₁ = a₃/a₂ = a₄/a₃, and so on. This ratio remains constant throughout the sequence.
Understanding the Common Ratio
The common ratio (r) determines the behavior of the sequence. When |r| > 1, the sequence grows exponentially. When 0 < |r| < 1, the sequence decreases toward zero. When r is negative, the sequence alternates between positive and negative values.
Sequence vs Series Distinction
A geometric sequence is the list of terms, while a geometric series is the sum of those terms. Understanding this distinction is crucial for applying the correct formulas and solving problems accurately.

Common Geometric Sequences

  • 2, 6, 18, 54, 162 (r = 3)
  • 100, 50, 25, 12.5, 6.25 (r = 0.5)
  • 1, -3, 9, -27, 81 (r = -3)

Mathematical Formulas and Derivations

  • nth Term Formula
  • Sum of First n Terms
  • Infinite Series Sum
The mathematical formulas for geometric sequences are derived from the fundamental property of constant ratios. These formulas allow us to find any term in the sequence or calculate the sum of multiple terms efficiently.
The nth Term Formula: aₙ = a₁ × r^(n-1)
This formula directly calculates any term in the sequence without finding all previous terms. The exponent (n-1) reflects that we multiply by the ratio (n-1) times to reach the nth term from the first term.
Sum Formula: Sₙ = a₁ × (1 - r^n) / (1 - r)
For r ≠ 1, this formula calculates the sum of the first n terms. When r = 1, the sequence is constant, and the sum is simply n × a₁. The derivation involves multiplying the sum by r and subtracting to eliminate most terms.
Infinite Series: S∞ = a₁ / (1 - r) for |r| < 1
When the absolute value of the common ratio is less than 1, the infinite series converges to a finite sum. This is because r^n approaches zero as n approaches infinity, making the sum formula limit to a₁/(1-r).

Formula Applications

  • a₅ = 2 × 3^(5-1) = 2 × 81 = 162
  • S₅ = 2 × (1 - 3⁵) / (1 - 3) = 2 × (-242) / (-2) = 242
  • S∞ = 1 / (1 - 0.5) = 2 (for a₁ = 1, r = 0.5)

Step-by-Step Calculator Usage Guide

  • Input Requirements
  • Calculation Types
  • Result Interpretation
Using the geometric sequence calculator effectively requires understanding the input parameters and choosing the appropriate calculation type for your specific problem. This guide walks through each step of the process.
Essential Input Parameters
The calculator requires three main inputs: the first term (a₁), the common ratio (r), and the number of terms (n). The first term can be any real number, the common ratio must be non-zero, and the number of terms must be a positive integer.
Selecting Calculation Types
Choose 'nth Term Only' to find a specific term, 'Sum of First n Terms' for partial sums, 'Infinite Series Sum' for convergent series (|r| < 1), or 'All Calculations' for comprehensive results including the sequence display.
Understanding Results
The calculator displays results with appropriate formulas and precision. For infinite sums, verify that |r| < 1 for convergence. Large exponents may result in very large or very small numbers, displayed in scientific notation when necessary.

Calculator Examples

  • Input: a₁ = 3, r = 2, n = 4 → Result: a₄ = 24
  • Input: a₁ = 10, r = 0.1, n = 3 → Sum: S₃ = 11.1
  • Input: a₁ = 8, r = 0.25 → Infinite sum: S∞ = 32/3

Real-World Applications

  • Financial Mathematics
  • Population Growth
  • Physics and Engineering
Geometric sequences appear frequently in real-world scenarios, particularly in situations involving exponential growth or decay. Understanding these applications helps connect mathematical theory to practical problem-solving.
Compound Interest and Investments
In finance, compound interest creates geometric sequences where each period's balance is the previous balance multiplied by (1 + interest rate). This application is fundamental in calculating investment growth, loan payments, and retirement planning.
Population Dynamics and Biology
Population growth under ideal conditions follows geometric patterns. Bacterial growth, viral spread models, and ecological population studies often use geometric sequences to predict future populations based on constant growth rates.
Physics and Engineering Applications
Radioactive decay, electrical circuits with capacitors, and signal processing all involve geometric sequences. Half-life calculations, RC circuit analysis, and digital signal processing rely on geometric sequence mathematics.

Practical Applications

  • $1000 at 5% annual interest: 1000, 1050, 1102.50, ...
  • Bacteria doubling hourly: 100, 200, 400, 800, 1600, ...
  • Radioactive decay (half-life): 1000, 500, 250, 125, 62.5, ...

Common Misconceptions and Problem-Solving Tips

  • Sequence vs Series Confusion
  • Convergence Criteria
  • Calculation Accuracy
Students often encounter specific challenges when working with geometric sequences. Understanding common misconceptions and developing effective problem-solving strategies can significantly improve mathematical proficiency in this area.
Avoiding Sequence and Series Mix-ups
The most common error is confusing sequences (lists of terms) with series (sums of terms). When asked for the 'fifth term,' use the nth term formula. When asked for the 'sum of five terms,' use the sum formula. Always read problems carefully to identify what's being requested.
Understanding Infinite Series Convergence
Infinite geometric series only converge when |r| < 1. If |r| ≥ 1, the series diverges (grows without bound). This is a critical concept that determines whether an infinite sum exists and is finite.
Maintaining Calculation Precision
Large exponents can lead to very large or very small numbers. Use scientific notation when appropriate, and be aware of calculator limitations. For theoretical work, express answers in exact form using fractions and exponents rather than decimal approximations.

Common Errors to Avoid

  • Wrong: 'Find the 5th term' → calculating sum
  • Wrong: Infinite sum with r = 2 → diverges
  • Wrong: 3^10 ≈ 59049 → use exact form when possible