Mathematics · Precalculus
Geometric Sequence Calculator
Find the nth term of a sequence with a constant multiplication ratio.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Term 5 applies the ratio 4 times.
- Compute 3^4 = 81.
- a_5 = 2 × 81 = 162.
Understand Geometric sequence
One idea, three depths
Choose how deeply to explain Geometric sequence
Geometric sequence: Find the nth term of a sequence with a constant multiplication ratio.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Geometric sequence to answer this question: find the nth term of a sequence with a constant multiplication ratio? Enter First term, Common ratio, Term number n; the calculator shows nth term. For example: Starting at 2 with ratio 3, term 5 is 2×3⁴=162. The answer tells you nth term.
Age 15Explain it to a 15-year-oldConnect it to the formula
Each move to the next term multiplies by r, so reaching term n applies the ratio n−1 times. The rule is aₙ = a₁rⁿ⁻¹. Its input values are First term, Common ratio, Term number n, and the main result is nth term. For example: Starting at 2 with ratio 3, term 5 is 2×3⁴=162.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated geometric sequence relation over the valid mixed integer and real-number domain stated below. The implemented relation is aₙ = a₁rⁿ⁻¹, evaluated from First term, Common ratio, Term number n to produce nth term. Each move to the next term multiplies by r, so reaching term n applies the ratio n−1 times. The exponent is n−1, not n.
Inputs and valid domain
- First term must be a finite real number.
- Common ratio must be a finite real number.
- Term number n must be an integer, at least 1.
Important boundary: The exponent is n−1, not n.
The formula
aₙ = a₁rⁿ⁻¹
How the calculator works through it
It substitutes First term, Common ratio, Term number n into the formula and exposes every numerical step above. The main output is nth term, accompanied by Ratio exponent.
Read the result correctly
The nth term is the direct answer to “find the nth term of a sequence with a constant multiplication ratio.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
Starting at 2 with ratio 3, term 5 is 2×3⁴=162.
Where this model stops being reliable
The exponent is n−1, not n.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Geometric sequence works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Geometric sequence uses aₙ = a₁rⁿ⁻¹. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Functions, domains and ranges
Domain and range language helps you identify which Geometric sequence inputs are valid and how the output behaves.
Review this foundation about 6 min
Optional enrichment
- Exponential growth and decay
Exponential models provide a useful extension when Geometric sequence is applied to multiplicative change.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read First term, Common ratio, Term number n.
- Evaluate the principal relationship: aₙ = a₁rⁿ⁻¹.
- Return nth term and check the domain conditions described above.
Python
from math import *
def geometric_sequence(a, r, n) -> float:
return (a * pow(r, (n - 1.0)))
assert abs(geometric_sequence(2, 3, 5) - 162) < 1e-6 * max(1.0, abs(162))
C
#include <assert.h>
#include <math.h>
double geometric_sequence(double a, double r, double n) {
return (a * pow(r, (n - 1.0)));
}
int main(void) {
const double expected = 162;
const double actual = geometric_sequence(2, 3, 5);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double geometric_sequence(double a, double r, double n) {
return (a * std::pow(r, (n - 1.0)));
}
int main() {
constexpr double expected = 162;
const double actual = geometric_sequence(2, 3, 5);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double geometric_sequence(double a, double r, double n)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern pow
global geometric_sequence
section .text
geometric_sequence:
push rbp
mov rbp, rsp
sub rsp, 64
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
mov rax, 0x3ff0000000000000
movq xmm0, rax
movsd [rbp-56], xmm0
movsd xmm0, [rbp-24]
subsd xmm0, [rbp-56]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-16]
movsd xmm1, [rbp-48]
call pow wrt ..plt
movsd [rbp-40], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-40]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = geometric_sequence(a, r, n)
result = (a * (r ^ (n - 1.0)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, r_, n_] := (a * (r ^ (n - 1.0)));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Geometric Sequence Calculator. MW SysArc Tools. https://math.mwsysarc.com/precalculus/geometric-sequence-calculator
MLA 9
MW SysArc. “Geometric Sequence Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/precalculus/geometric-sequence-calculator. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Geometric Sequence Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/precalculus/geometric-sequence-calculator.
Harvard
MW SysArc (2026) ‘Geometric Sequence Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/precalculus/geometric-sequence-calculator (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_geometric_sequence_2026,
author = {{MW SysArc}},
title = {Geometric Sequence Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/precalculus/geometric-sequence-calculator},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Geometric Sequence Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/precalculus/geometric-sequence-calculator
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Geometric sequence do?
Find the nth term of a sequence with a constant multiplication ratio.
How does the Geometric sequence work?
The calculator applies aₙ = a₁rⁿ⁻¹. Each move to the next term multiplies by r, so reaching term n applies the ratio n−1 times.
What can I learn from the Geometric sequence?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .