Geometric Sequence Calculator

Geometric Sequence Calculator

Calculating geometric sequence…

Geometric Sequence Results

n-th term (an):
Sum of first n terms (Sn):
Sum to infinity (|r|<1):
Formulas Used:
n-th term: an = a1 × rn-1
Sum of n terms: Sn = a1 × (1 – rn) / (1 – r), r ≠ 1
Sum to infinity: S∞ = a1 / (1 – r), |r| < 1

Geometric sequences appear everywhere—finance, physics, computer science, and even nature. A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio.

Our Geometric Sequence Calculator simplifies the process of finding terms, sums, and even the sum to infinity. Whether you’re a student, teacher, or professional, this tool saves time, reduces calculation errors, and provides quick results with clear formulas.

How to Use the Geometric Sequence Calculator

Using the tool is simple and requires just three inputs.

Step-by-Step Instructions

  1. Enter the First Term (a₁)
    Type the first term of your sequence in the input box.
  2. Enter the Common Ratio (r)
    This is the multiplier between terms. It can be positive, negative, or a fraction.
  3. Enter the Number of Terms (n)
    Choose how many terms you want to calculate. You can enter up to 100.
  4. Click “Calculate”
    The tool will display a progress animation and then present:
    • The sequence (up to the first 12 terms)
    • The n-th term
    • The sum of the first n terms
    • The sum to infinity (if |r| < 1)
  5. Optional – Copy or Share
    Use the Copy Results or Share Results buttons to keep or send your results.

Example: Calculating a Geometric Sequence

Problem: Find the first 6 terms of a geometric sequence where:

  • a₁ = 3
  • r = 2
  • n = 6

Steps in the Calculator:

  1. First Term = 3
  2. Common Ratio = 2
  3. Number of Terms = 6
  4. Click Calculate

Results:

  • Sequence: 3, 6, 12, 24, 48, 96
  • n-th term (6th term): 96
  • Sum of first 6 terms: 189
  • Sum to infinity: Not convergent (since |r| ≥ 1)

Benefits of Using the Calculator

  • Saves Time – No need for manual, repetitive calculations.
  • Reduces Errors – Eliminates mistakes common in hand calculations.
  • Educational Value – Shows formulas used, reinforcing learning.
  • Multi-Use – Works for any valid a₁, r, and n values, including fractions and decimals.
  • Portable – Copy or share results instantly.

Features at a Glance

  • Instant Results with animated calculation progress.
  • Sequence Display for up to 12 terms (with ellipsis for longer sequences).
  • Formula Display for learning and verification.
  • Copy & Share functionality for convenience.
  • Handles Complex Inputs – Negative ratios, fractions, large term counts.

Use Cases

  • Mathematics Homework – Quickly solve textbook problems.
  • Financial Projections – Model compound interest or depreciation.
  • Computer Science – Analyze algorithm complexity in geometric growth scenarios.
  • Engineering – Calculate repetitive scaling patterns.
  • Research – Model population growth, decay, or resource use.

Pro Tips for Accurate Results

  • Always double-check that n is a whole number greater than zero.
  • If |r| ≥ 1, the sum to infinity will not converge.
  • For very large n, be aware that numbers may become extremely large or small.
  • Use decimal form for fractions (e.g., 0.5 instead of 1/2) for quicker input.

Frequently Asked Questions (FAQ)

1. What is a geometric sequence?
A sequence where each term is obtained by multiplying the previous term by a constant ratio.

2. What does the calculator require?
First term (a₁), common ratio (r), and number of terms (n).

3. Can I use negative ratios?
Yes, the tool supports both positive and negative ratios.

4. What happens if r = 1?
The sequence is constant, and the sum of n terms is simply a₁ × n.

5. Can it handle decimal values?
Yes, you can input decimal first terms and ratios.

6. What is the “sum to infinity”?
The total sum of all terms if |r| < 1, calculated as a₁ / (1 − r).

7. Why does “sum to infinity” sometimes say “Not convergent”?
Because if |r| ≥ 1, the sequence grows without bound and doesn’t converge.

8. Is there a limit to n?
Yes, the tool allows up to 100 terms.

9. Why only show the first 12 terms?
To keep results readable; larger n still calculates correct sums and nth term.

10. Can I copy the results?
Yes, click Copy Results to save all data to your clipboard.

11. How does sharing work?
It uses your device’s share function or copies text for manual sharing.

12. Can I use it for compound interest calculations?
Yes, since compound interest follows a geometric sequence.

13. What if my ratio is a fraction?
Enter it as a decimal (e.g., 0.25 for 1/4).

14. Will it work on mobile devices?
Yes, it’s fully responsive.

15. What’s the maximum value it can display?
It depends on your device’s number precision, but it handles large values well.

16. Can I reset my inputs?
Yes, click the Reset button to start over.

17. Does it show the formula used?
Yes, the exact formulas for nth term, sum of n terms, and sum to infinity are displayed.

18. Why does my sequence look like it’s alternating?
A negative ratio causes alternating positive and negative terms.

19. Can it calculate backwards?
Yes, if your ratio is between −1 and 0, terms decrease in magnitude.

20. Is it free to use?
Yes, the calculator is completely free.

This article gives users everything they need to confidently use the Geometric Sequence Calculator for learning, problem-solving, and professional applications.