Algebra • Sequences & Series Flagship

Geometric Sequence Calculator

Solve geometric progressions (GP) for any target $n$-th term ($a_n$), common ratio ($r$), finite partial sum ($S_n$), or sum to infinity ($S_\infty$). Generates explicit formulas, discrete SVG exponential plots, and cumulative sum tables.

Reviewed by Sanjay Samanta & Academic Review Board

Last Updated: September 2026

Verified Accurate: Infinite Series Convergence Theorems

Sequence & Series Solver

Calculate nth terms, finite partial sums, infinite series convergence, and explicit formulas.

Input Mode:

Quick Presets:

First Term (a₁)

Common Difference (d)

Target Term Index (n)

Arithmetic Progression Solution (AP)

Calculated nth-term value, partial series sum, and closed-form algebraic formulas.

Nth Term (a₁₀) Target

a₁₀ = 3 + 9(4)

Sum of First n Terms (S₁₀) Series Sum

210

S₁₀ = (10/2)(3 + 39)

Common Difference (d) Step

Constant delta

Initial Value (a₁) Base

Starting term at n=1

Explicit Formula

a_n = 3 + 4(n - 1) = 4n - 1

Recursive Formula

a₁ = 3, a_n = a_{n-1} + 4

Discrete Sequence Progression Chart

First 8 Terms

Sequence Points $(k, a_k)$

Linear Progression (+d)

Sequence Terms Table

Cumulative Sum ($S_k$)

n	Term (a_n)	Sum (S_n)

Direct Answer & Overview

Verified Educational Guide

Geometric Sequence Overview & Direct Answer

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the preceding term by a fixed non-zero constant called the common ratio (r). The nth term is given by a_n = a_1 · r^(n - 1). For common ratio |r| < 1, the infinite series converges to S_∞ = a_1 / (1 - r).

Primary Mathematical Formula Universal Explicit Term, Finite Sum & Infinite Series Convergence

Standard Equation

ƒ(x)

Q.E.D.

a_n = a_1 \cdot r^{n-1}, \quad S_n = \frac{a_1(1 - r^n)}{1 - r}, \quad S_\infty = \frac{a_1}{1 - r} \quad (|r| < 1)

Finite sum requires r ≠ 1. Infinite series converges if and only if |r| < 1.

Exact Formula

Input Parameters

Required

First Term (a₁): The initial real non-zero starting element of the geometric sequence.

Common Ratio (r): The constant scalar multiplier applied to generate each successive term.

Target Term Index (n): The positive integer index (1, 2, 3...) of the desired sequence position.

Expected Outputs

Calculated

Target Nth Term (a_n): The exact value at index n, computed via a_n = a_1 · r^(n - 1).

Finite Series Sum (S_n): The cumulative partial sum of the first n terms.

Infinite Series Sum (S_∞): The convergent limit of the infinite series if |r| < 1, or divergence status.

Explicit and Recursive Formulas: The closed-form analytical equations defining the progression.

Worked Numerical Example

Instant Verification

Find the 5th term and sum of first 5 terms for sequence 2, 6, 18, 54, ...

→ Step 1: Identify initial term a_1 = 2 and common ratio r = 6/2 = 3. Step 2: Evaluate 5th term: a_5 = 2 · 3^(5 - 1) = 2 · 81 = 162. Step 3: Evaluate finite sum: S_5 = 2(1 - 3^5)/(1 - 3) = 2(-242)/(-2) = 242.

a_5 = 162, S_5 = 242

Geometric Progression Foundations & Common Ratio

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed scalar factor $r$, known as the common ratio:

r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \dots = \frac{a_n}{a_{n-1}} \quad (a_k \neq 0)

Geometric progressions differ fundamentally from additive progressions. In additive recurrences such as the Fibonacci Sequence Generator, terms are produced by summing prior terms. In a geometric progression, the change is purely multiplicative, yielding exponential growth or decay curves analyzed in our Exponential Growth Calculator.

Exponential Growth (r > 1)

Magnitudes grow exponentially without bound (e.g. $2, 6, 18, 54\dots$ with $r = 3$).

Exponential Decay (0 < |r| < 1)

Magnitudes shrink progressively toward zero (e.g. $100, 50, 25, 12.5\dots$ with $r = 0.5$).

Alternating Sign (r < 0)

Terms alternate between positive and negative (e.g. $4, -8, 16, -32\dots$ with $r = -2$).

Derivation of the Nth-Term Formula

By inspecting how each consecutive term compounds multiplicatively from the initial value $a_1$:

Term 1 ($n=1$): $a_1 = a_1 \cdot r^0$

Term 2 ($n=2$): $a_2 = a_1 \cdot r^1$

Term 3 ($n=3$): $a_3 = a_2 \cdot r = (a_1 \cdot r) \cdot r = a_1 \cdot r^2$

Term 4 ($n=4$): $a_4 = a_3 \cdot r = (a_1 \cdot r^2) \cdot r = a_1 \cdot r^3$

Term $n$: $a_n = a_1 \cdot r^{n - 1}$

To reach position $n$ starting from position 1, the ratio $r$ is applied as a multiplicative factor exactly $n - 1$ times. When working with non-integer power exponents, explore our Fractional Exponent Calculator.

Finite Geometric Series Sum Derivation

The sum of the first $n$ terms $S_n = a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1}$ is derived using algebraic multiplication and telescoping subtraction:

Telescoping Series Proof

Write the finite series sum equation:
S_n = a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1} — (Equation 1)
Multiply the entire equation by the common ratio $r$:
r S_n = a_1 r + a_1 r^2 + a_1 r^3 + \dots + a_1 r^n — (Equation 2)
Subtract Equation 2 from Equation 1. All intermediate terms from $a_1 r$ through $a_1 r^{n-1}$ cancel out:
S_n - r S_n = a_1 - a_1 r^n \implies S_n(1 - r) = a_1(1 - r^n)
Divide both sides by $(1 - r)$ (assuming $r \neq 1$):
S_n = \frac{a_1(1 - r^n)}{1 - r} = \frac{a_1(r^n - 1)}{r - 1}

Infinite Geometric Series & Convergence Criterion

An infinite geometric series has an infinite number of terms ($n \to \infty$). Evaluating its limit using the finite sum formula:

S_\infty = \lim_{n \to \infty} \frac{a_1(1 - r^n)}{1 - r}

Convergent Case ($|r| < 1$)

When $|r| < 1$, $\lim_{n \to \infty} r^n = 0$. The sum converges to a finite value:

S_\infty = \frac{a_1}{1 - r}

Divergent Case ($|r| \ge 1$)

When $|r| \ge 1$, $r^n$ grows infinitely or oscillates without settling. The infinite sum diverges and has no finite sum.

Real-World Applications in Finance, Physics & Fractals

Compound Interest & Annuities

Annual compound balance growth represents a geometric sequence with ratio $r = 1 + i$. Retirement annuity payout streams are calculated as finite geometric series sums.

Radioactive Decay & Half-Life

Nuclear isotopes decay by halving their remaining mass every half-life interval ($r = 0.5$). Remaining activity follows an exact decaying geometric sequence.

Fractal Geometry (Koch Snowflake)

Fractals like the Koch Snowflake or Sierpinski Triangle generate infinite perimeter and finite enclosed area by summing infinite geometric series ($r = 4/9$).

Graded Step-by-Step Numerical Solutions

Example 1 • Standard Finite Progression Basic Tier

Find the 6th term and sum of first 6 terms for $a_1 = 4, r = 3$.

1. Compute $a_6$: $a_6 = 4 \cdot 3^{6 - 1} = 4 \cdot 3^5 = 4 \cdot 243 = 972$.

2. Compute $S_6$: $S_6 = \frac{4(1 - 3^6)}{1 - 3} = \frac{4(1 - 729)}{-2} = \frac{4(-728)}{-2} = 1456$.

Example 2 • Convergent Infinite Geometric Series Intermediate Tier

Evaluate the sum of the infinite series $18 + 6 + 2 + \frac{2}{3} + \dots$

1. Identify parameters: $a_1 = 18$, $r = \frac{6}{18} = \frac{1}{3}$.

2. Verify convergence: $|r| = \left|\frac{1}{3}\right| < 1$, so the series converges.

3. Calculate $S_\infty$: $S_\infty = \frac{18}{1 - \frac{1}{3}} = \frac{18}{\frac{2}{3}} = 18 \times \frac{3}{2} = 27$.

Common Calculation Pitfalls & Mistake Avoidance

Pitfall 1: Applying Infinite Series Formula When $|r| \ge 1$

The formula $S_\infty = \frac{a_1}{1-r}$ is mathematically valid only when $|r| < 1$. Applying it to a divergent sequence (e.g. $a_1 = 2, r = 3 \implies \frac{2}{1-3} = -1$) produces nonsensical negative numbers for an infinitely growing positive sum.

Pitfall 2: Forgetting Parentheses Around Negative Bases in Exponentiation

When $r < 0$, $(-r)^n \neq -r^n$. For example, $(-2)^4 = +16$, whereas $-2^4 = -16$. Always enclose negative ratios in parentheses when evaluating powers on a calculator.

Fact-Checked & Verified • Computational Accuracy Standards

Updated September 2026 • Editorial Policy

Authored By

Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By

Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

What is the formula for the nth term of a geometric sequence?

The explicit formula is a_n = a_1 · r^(n − 1), where a_1 is the first term, r is the common ratio, and n is the position index of the term.

How do you find the sum of a finite geometric series (S_n)?

For common ratio r ≠ 1, the finite sum is S_n = a_1(1 − r^n) / (1 − r). If r = 1, the sum is simply S_n = n · a_1.

When does an infinite geometric series converge to a finite sum?

An infinite geometric series converges if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). The sum to infinity is given by S_∞ = a_1 / (1 − r).

How do you find the common ratio (r) given two arbitrary terms?

Given term a_p at index p and term a_q at index q (with q > p), the ratio satisfies r^(q − p) = a_q / a_p, which gives r = (a_q / a_p)^(1 / (q − p)). The initial term is then a_1 = a_p / r^(p − 1).

What happens when the common ratio (r) is negative?

When r < 0, the terms alternate in sign (oscillating between positive and negative values). For example, with a_1 = 4 and r = −2, the sequence is 4, −8, 16, −32, 64, −128.