Last updated: July 31, 2026
Geometric Sequence Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
aₙ = a₁·r^(n−1); Sₙ = a₁·(1−rⁿ)/(1−r) for r ≠ 1, otherwise Sₙ = n·a₁
Where:
- firstTerm=First Term (a₁)
- ratio=Common Ratio (r)
- n=Term Number (n)
Worked Examples
Growing sequence
Find the 6th term and sum when a₁=3 and r=2.
- 1Apply aₙ = a₁·r^(n−1)
- 2Compute 3·2^5 for the 6th term
- 3Use geometric-sum formula for first 6 terms
Constant sequence
Use ratio 1 to get repeated terms.
- 1All terms equal first term when r=1
- 2a₄ stays 5
- 3Sum is n·a₁ = 4·5
Alternating signs
Use a negative ratio.
- 1Signs alternate with negative ratio
- 2Compute a₅ from exponent n−1
- 3Add first 5 terms for S₅
Introduction
A geometric sequence multiplies each term by the same ratio. This calculator returns the requested term and the cumulative sum for fast sequence analysis.
Formula Overview
Geometric Sequence Calculator uses a deterministic math model based on the calculator logic in calculation.ts.
aₙ = a₁·r^(n−1); Sₙ = a₁·(1−rⁿ)/(1−r) for r ≠ 1, otherwise Sₙ = n·a₁
Inputs are validated before computation
Outputs are rounded consistently for readable results
Invalid values return safe fallback outputs
Input Guide
Use each input key exactly as defined below to match calculator wiring and test coverage.
- 1
firstTerm: First Term (a₁)
- 2
ratio: Common Ratio (r)
- 3
n: Term Number (n)
Output Guide
These output IDs map directly to the return object keys in calculation.ts.
- 1
nthTerm: Value of the requested term aₙ
- 2
sumOfNTerms: Total of the first n terms in the sequence
Validation Rules
The calculator checks for finite numeric values and applies domain constraints before solving.
Required fields must be present
Domain limits are enforced (for example positive lengths or valid operators)
Invalid or non-finite entries resolve to safe defaults
Use examples to verify expected behavior quickly
How to Use
Enter inputs, run calculation, and interpret the primary output first before reviewing supporting values.
Provide all required inputs
Click calculate
Read primary output first
Use secondary outputs for deeper analysis
Practical Uses
This calculator supports classroom work, engineering checks, and fast verification tasks.
Homework and exam preparation
Design and geometry validation
Spreadsheet cross-checking
Quick scenario analysis
FAQs
What does the Geometric Sequence Calculator compute?
It computes n-th Term, Sum of First n Terms from validated input values.
Which inputs are required?
Required inputs are: First Term (a₁), Common Ratio (r), Term Number (n).
How are invalid values handled?
If inputs are invalid or out of domain, the calculator returns safe default outputs instead of invalid math.
Are results rounded?
Yes. Numeric values are rounded in calculation.ts (typically to six decimal places).
Can I use negative or decimal values?
Decimals are accepted where mathematically valid; sign/domain constraints are enforced by the calculator logic.
How can I verify results?
Use the worked examples and compare each output key with manual calculations or trusted references.