Sum of Geometric Series Calculator: Formulas, Examples & Free Tool

Quick calculation needed? Use our free Sum of Series Calculator to find geometric series sums instantly—works for both finite and infinite series.

What Is a Geometric Series? #

A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio ($r$).

For example: $2,6,18,54,162,...$

Here, each term is 3 times the previous one, so $r=3$.

Geometric series appear everywhere:

• Compound interest calculations
• Population growth models
• Physics (bouncing balls, radioactive decay)
• Computer science (algorithm analysis)
• Music theory (frequency ratios)

Knowing how to calculate their sums—and having a reliable calculator to verify your work—is essential for students and professionals alike.

Table of Contents #

• The Geometric Series Formulas
• Using a Geometric Series Calculator
• Finite Geometric Series Examples
• Infinite Geometric Series Examples
• Common Mistakes to Avoid
• Frequently Asked Questions

The Geometric Series Formulas #

Finite Geometric Series #

For a series with first term $a$, common ratio $r$, and $n$ terms:

$S_n=a\cdot \frac{1-r^n}{1-r}(r\ne 1)$

If $r=1$, every term equals $a$, so $S_n=n\cdot a$.

Infinite Geometric Series #

When $|r|<1$, the series converges to a finite sum:

$S_{\infty }=\frac{a}{1-r}(|r|<1)$

When $|r|\ge 1$, the series diverges—it doesn't have a finite sum.

Using a Geometric Series Calculator #

A geometric series calculator saves time and eliminates arithmetic errors. Here's how to use one effectively:

What You Need to Input #

Step-by-Step Process #

1. Identify the first term — What's the starting value of your series?

2. Find the common ratio — Divide any term by the previous term: $r=\frac{a_2}{a_1}$

3. Determine if it's finite or infinite — Do you have a specific number of terms, or does it continue forever?

4. Enter values into the calculator — Use our Sum of Series Calculator for instant results.

5. Interpret the result — For infinite series, check if $|r|<1$ to confirm convergence.

Finite Geometric Series Examples #

Example 1: Simple Finite Series #

Problem: Find the sum of $3+6+12+24+48+96$.

Solution:

• First term: $a=3$
• Common ratio: $r=\frac{6}{3}=2$
• Number of terms: $n=6$

$S_6=3\cdot \frac{1-2^6}{1-2}=3\cdot \frac{1-64}{-1}=3\cdot 63=189$

Answer: The sum is 189.

Example 2: Compound Interest #

Problem: You invest $1,000 at 5% annual interest. What's the total value after 10 years with compound interest?

This is a geometric series where each year's value is 1.05 times the previous year.

Solution:

• First term: $a=1000$
• Common ratio: $r=1.05$
• Number of terms: $n=10$

The value after 10 years (not the sum, but the 10th term): $a_{10}=1000\cdot (1.05{)}^{10}\approx 1628.89$

If you wanted the sum of all yearly values (less common), you'd use the series formula.

Example 3: Decreasing Series #

Problem: Find the sum: $100+50+25+12.5+6.25$

Solution:

• First term: $a=100$
• Common ratio: $r=0.5$
• Number of terms: $n=5$

$S_5=100\cdot \frac{1-(0.5{)}^5}{1-0.5}=100\cdot \frac{1-0.03125}{0.5}=100\cdot 1.9375=193.75$

Answer: The sum is 193.75.

(Need help with other series types? Check out our guide on How to Sum a Series.)

Infinite Geometric Series Examples #

Example 1: Classic Convergent Series #

Problem: Find the sum of $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$

Solution:

• First term: $a=1$
• Common ratio: $r=\frac{1}{2}$
• Since $|r|=0.5<1$, the series converges.

$S_{\infty }=\frac{1}{1-0.5}=\frac{1}{0.5}=2$

Answer: The infinite sum equals 2.

Example 2: Repeating Decimal #

Problem: Express $\mathrm{0.333...}$ as a fraction using geometric series.

Solution: $\mathrm{0.333...}=0.3+0.03+0.003+...$

• First term: $a=0.3$
• Common ratio: $r=0.1$

$S_{\infty }=\frac{0.3}{1-0.1}=\frac{0.3}{0.9}=\frac{1}{3}$

Answer: $\mathrm{0.333...}=\frac{1}{3}$

Example 3: Bouncing Ball #

Problem: A ball is dropped from 10 meters and bounces back to 60% of its previous height each time. What's the total distance traveled?

Solution:

The ball falls 10m, then bounces up 6m and falls 6m, then bounces up 3.6m and falls 3.6m, etc.

Total distance = Initial drop + 2 × (sum of bounce heights)

Bounce heights form a geometric series: $6+3.6+2.16+...$

• First term: $a=6$
• Common ratio: $r=0.6$

$S_{\infty }=\frac{6}{1-0.6}=\frac{6}{0.4}=15$

Total distance = $10+2(15)=40$ meters

Answer: The ball travels 40 meters total.

(For more infinite series calculations, try our Infinite Sum Calculator.)

Common Mistakes to Avoid #

When to Use a Calculator #

Use a calculator when:

• Dealing with large $n$ values (computing $r^{50}$ by hand is tedious)
• Working with non-integer ratios
• Checking homework answers
• Under time pressure

Calculate manually when:

• Learning the concept
• Taking exams without calculator access
• Working with simple values ($r=2$, small $n$)

The best strategy: understand the formulas first, then use calculators to work efficiently.

Frequently Asked Questions #

Q: How do I find the common ratio?

Divide any term by the previous term: $r=\frac{a_{n+1}}{a_n}$. For the series $4,12,36,...$, the ratio is $r=\frac{12}{4}=3$.

Q: What if the common ratio is negative?

The formula still works! A negative ratio means terms alternate between positive and negative. For example: $1,-2,4,-8,...$ has $r=-2$.

Q: Can a geometric series have $r=1$?

Technically yes, but it's just the same number repeated: $a,a,a,...$. The sum of $n$ terms is simply $S_n=n\cdot a$. The standard formula doesn't apply since it would divide by zero.

Q: How do I know if an infinite series converges?

Check the absolute value of the common ratio. If $|r|<1$, it converges. If $|r|\ge 1$, it diverges (no finite sum).

Q: What's the difference between geometric and arithmetic series?

In an arithmetic series, you add a constant difference between terms ($2,5,8,11,...$). In a geometric series, you multiply by a constant ratio ($2,6,18,54,...$).

Related Calculations #

Geometric series are just one type of summation. Explore these related tools:

• Summation Calculator — General-purpose sum calculator
• Sum of Series Calculator — Multiple series types
• Infinite Sum Calculator — Convergent infinite series
• Average Calculator — Mean, median, mode calculations
• Riemann Sum Calculator — Integral approximations

Final Thoughts #

Geometric series calculations are straightforward once you know the formulas—but even small arithmetic errors can throw off your answer. That's where a calculator becomes invaluable.

Whether you're computing compound interest, analyzing convergence, or solving physics problems, bookmark our Sum of Series Calculator for quick, accurate results.

Master the formulas, understand when series converge, and let the calculator handle the heavy lifting. Your math homework will thank you.