Geometric Sequence Calculator
Find the nth Term, Partial Sum, and Infinite Sum
$$ a_n = a_1 \cdot r^{n-1} $$
First Term ($a_1$)
Common Ratio ($r$)
Term Index ($n$)
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nth Term ($a_n$)
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Sum ($S_n$)
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Sequence Graph
Calculation Steps
First $n$ Terms
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By Prof. David Anderson
Professor of Applied Mathematics | 20+ Years Teaching Experience
“Unlike Arithmetic Sequences which simply add up, Geometric Sequences explode with the power of multiplication. This is the math of viruses spreading, investments doubling, and nuclei splitting. Students often get tripped up by the ‘Infinite Sum’ concept—how can you add forever and get a finite number? I built this Geometric Sequence Calculator to not only do the math but to visualize this convergence for you.”
The Professor’s Guide to Using a Geometric Sequence Calculator: Formulas, Series, and Sums
A Complete Handbook on Exponential Growth, Convergence, and Partial Sums
⚡ Key Takeaways for Students
- A Geometric Sequence (or Geometric Progression) multiplies by a constant Common Ratio ($r$) at each step.
- Nth Term Formula: $a_n = a_1 \cdot r^{n-1}$. Used by our calculator to find any specific value in the list.
- Sum of Geometric Sequence Calculator: Computes the partial sum ($S_n$) for loans or finite growth models.
- Infinite Geometric Series Calculator: Computes $S_\infty$ only if $|r| < 1$. It adds forever but approaches a limit.
Welcome to the definitive guide on Geometric Progressions (G.P.). While arithmetic sequences are linear (like climbing a ladder), geometric sequences are exponential (like a rocket taking off). Whether you need a Geometric Series Calculator for calculus or a simple Geometric Sequence Solver for algebra, understanding the underlying formulas is key.
Our Geometric Sequence Calculator above is designed to handle the three critical tasks: finding the specific term ($a_n$), calculating the partial sum ($S_n$), and determining if the infinite geometric series converges ($S_\infty$).
1. Understanding the Geometric Sequence Formula and Variables
Every geometric sequence is defined by just two numbers: the start and the multiplier. To use the calculator effectively, you must identify:
| Symbol | Name | Definition | Example (3, 6, 12…) |
|---|---|---|---|
| $a_1$ | First Term | The starting value of the sequence. | $3$ |
| $r$ | Common Ratio | The factor we multiply by ($r = a_2 / a_1$). | $6 / 3 = 2$ |
| $n$ | Term Position | Which step we are on (1st, 2nd, 10th…). | $n=10$ |
2. Calculating the Nth Term of a Geometric Sequence
How do we find the 100th term without writing out the first 99? We use the Explicit Formula for Geometric Sequence.
Explicit Formula
$$ a_n = a_1 \cdot r^{n-1} $$
Why $n-1$? Because to get to the 1st term, we multiply by $r$ zero times. To get to the 2nd term, we multiply once. To get to the $n$th term, we multiply $n-1$ times. This is the core logic behind our Nth Term Calculator.
3. Sum of Geometric Sequence Calculator: Finite and Infinite
A “Series” is simply the sum of a sequence. The formula changes slightly depending on your common ratio $r$.
Finite Geometric Series ($S_n$)
Use this when adding a specific number of terms (e.g., “Sum of the first 10 terms”).
$$ S_n = \frac{a_1(1-r^n)}{1-r} \quad \text{or} \quad S_n = \frac{a_1(r^n-1)}{r-1} $$
Pro Tip: Our Geometric Sequence Sum Calculator automatically chooses the second version when $r > 1$ to avoid negative numbers in the fraction.
Infinite Geometric Series ($S_\infty$)
This is where math gets magical. If the sequence is shrinking (decaying), the sum approaches a limit. This is often called a Convergent Geometric Series.
Convergence Rule: An infinite sum only exists if $|r| < 1$ (i.e., $-1 < r < 1$). If $|r| \ge 1$, the sum explodes to infinity (Diverges).
$$ S_\infty = \frac{a_1}{1-r} $$
4. Geometric Progression in the Real World
Application: The Bouncing Ball
A ball is dropped from 10 meters ($a_1=10$). Each time it hits the ground, it bounces back up to 80% of its previous height ($r=0.8$).
Total Distance Traveled?
This models an Infinite Geometric Series. Since $r=0.8 < 1$, the ball travels a finite distance even though it bounces infinitely many times.
$$ S_\infty = \frac{10}{1 – 0.8} = \frac{10}{0.2} = 50 \text{ meters} $$
5. Arithmetic vs. Geometric: What’s the difference?
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Operation | Addition (+) | Multiplication ($\times$) |
| Constant | Common Difference ($d$) | Common Ratio ($r$) |
| Graph Shape | Linear (Straight Line) | Exponential (Curve) |
| Example | 2, 4, 6, 8 ($+2$) | 2, 4, 8, 16 ($\times 2$) |
6. Frequently Asked Questions (FAQ)
How do I find the common ratio?
To find the common ratio $r$, simply divide any term by the previous term: $r = a_2 / a_1$. If the ratio is consistent across all terms, it is a geometric sequence.
Can r equal 1?
Technically yes, but it’s boring. The sequence would be constant (5, 5, 5…). The sum formulas divide by $(1-r)$, so if $r=1$, you would divide by zero. In this case, $S_n = n \times a_1$.
What is Zeno’s Paradox?
It’s an ancient philosophical problem solved by geometric series. “To walk across a room, you must first go half way ($1/2$), then half the remaining distance ($1/4$), then half again ($1/8$)…” Zeno argued you’d never arrive. Calculus proves that $\sum (1/2)^n = 1$. You do arrive!
References & Further Reading
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning. (Chapter 11: Sequences and Series).
- Larson, R. (2021). Precalculus (11th ed.). Cengage Learning. (Chapter 9: Sequences, Series, and Probability).
- Khan Academy. “Geometric sequences and series.” Start Learning
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Whether you are solving for homework or calculating compound interest, precision matters. Use our free Geometric Sequence Calculator to get instant steps, sums, and visual graphs.
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