Sum of Geometric Series Calculator & Formula

Sum of Geometric Series Calculator

Series Type: Finite Series Infinite Series

First Term (a):

The initial term of the series.

Common Ratio (r):

The constant factor between successive terms. For infinite series, |r| must be less than 1.

Number of Terms (n):

The total count of terms in the finite series (must be a positive integer).

Chart showing the value of each term and the cumulative sum of the series.

Term (k)	Term Value (a*r^(k-1))	Cumulative Sum (S_k)

Table detailing the first few terms and their cumulative sum.

What is the Sum of a Geometric Series?

The sum of a geometric series is the total obtained by adding up the terms of a geometric sequence. A geometric sequence (or progression) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). The Sum of Geometric Series Calculator helps you find this sum for both finite and infinite series.

For example, the sequence 2, 4, 8, 16, 32 is a geometric sequence with a first term (a) of 2 and a common ratio (r) of 2. The sum of these 5 terms is 2 + 4 + 8 + 16 + 32 = 62.

Anyone dealing with exponential growth or decay, such as in finance (compound interest), physics (decay processes), or computer science (algorithms), might use a Sum of Geometric Series Calculator. A common misconception is that all infinite geometric series have an infinite sum; however, if the absolute value of the common ratio is less than 1, the infinite series converges to a finite sum.

Sum of Geometric Series Formula and Mathematical Explanation

There are different formulas for the sum of a finite geometric series and an infinite geometric series.

Finite Geometric Series

A finite geometric series has a specific number of terms (n). The sum (S_n) is given by:

If r ≠ 1: S_n = a(1 - rⁿ) / (1 - r)

If r = 1: S_n = n * a

Where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.

Infinite Geometric Series

An infinite geometric series converges to a finite sum only if the absolute value of the common ratio |r| < 1. The sum (S_∞) is given by:

If |r| < 1: S_∞ = a / (1 – r)

If |r| ≥ 1, the series diverges and does not have a finite sum (unless a=0).

Variables Table

Variable	Meaning	Unit	Typical Range
a	First Term	Dimensionless or units of the term	Any real number
r	Common Ratio	Dimensionless	Any real number (but \|r\|<1 for infinite sum convergence)
n	Number of Terms	Dimensionless	Positive integer (for finite series)
S_n	Sum of first n terms	Dimensionless or units of the term	Depends on a, r, n
S_∞	Sum of infinite terms	Dimensionless or units of the term	Depends on a, r (only if \|r\|<1)

Practical Examples (Real-World Use Cases)

Example 1: Savings Plan

Suppose you save $100 in the first month and decide to increase your savings by 5% each month for 12 months. This is a finite geometric series with a=100, r=1.05, and n=12. Using the Sum of Geometric Series Calculator or formula: S₁₂ = 100 * (1 – 1.05¹²) / (1 – 1.05) ≈ 100 * (1 – 1.795856) / (-0.05) ≈ $1591.71. Your total savings after 12 months would be about $1591.71.

Example 2: Bouncing Ball

A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 70% of its previous height. The total distance the ball travels downwards is 10 + 10*0.7 + 10*0.7² + … This is an infinite geometric series with a=10 and r=0.7. Since |0.7| < 1, the sum converges. S_∞ (downwards) = 10 / (1 – 0.7) = 10 / 0.3 ≈ 33.33 meters. The total distance upwards is similar but starts after the first drop, so it's 7 + 7*0.7 + …, which is 7 / (1-0.7) ≈ 23.33 meters. Total distance = 33.33 + 23.33 = 56.66 meters. The Sum of Geometric Series Calculator can quickly find the sum for infinite series like this.

How to Use This Sum of Geometric Series Calculator

Select Series Type: Choose 'Finite Series' if you know the number of terms, or 'Infinite Series' if the series goes on forever.
Enter First Term (a): Input the initial value of your series.
Enter Common Ratio (r): Input the constant multiplier between terms. For infinite series, ensure |r| < 1 for a finite sum.
Enter Number of Terms (n): If you selected 'Finite Series', enter the total number of terms. This must be a positive integer.
View Results: The calculator automatically updates the sum, intermediate values, formula used, table, and chart as you enter the values.
Interpret Results: The 'Primary Result' shows the sum (S_n or S_∞). Intermediate results and the formula provide context. The table and chart visualize the series.

Use the Sum of Geometric Series Calculator to quickly find sums without manual calculation, especially useful for long series or when exploring different 'r' values.

Key Factors That Affect the Sum of a Geometric Series

First Term (a): A larger first term directly scales the sum proportionally.
Common Ratio (r): This is the most critical factor.
- If |r| < 1, the terms decrease, and an infinite series converges. The closer |r| is to 1, the larger the sum.
- If |r| > 1, the terms increase, and both finite and infinite series (except if a=0) grow rapidly (diverge).
- If r = 1, it's a constant sequence, and the sum is n*a (finite).
- If r is negative, terms alternate in sign.
Number of Terms (n) (for finite series): More terms generally lead to a larger magnitude of the sum, especially if |r| > 1.
Convergence Condition (|r| < 1 for infinite series): If this condition is not met for an infinite series, the sum is undefined or infinite (the series diverges). Our Sum of Geometric Series Calculator will indicate this.
Sign of 'a' and 'r': The signs of the first term and common ratio determine the signs of the individual terms and thus the overall sum.
Magnitude of 'r' relative to 1: Whether |r| is less than, equal to, or greater than 1 drastically changes the behavior of the sum.

Frequently Asked Questions (FAQ)

What is a geometric series?: A geometric series is the sum of terms in a geometric sequence, where each term after the first is found by multiplying the previous one by a fixed common ratio.
When does an infinite geometric series have a finite sum?: An infinite geometric series has a finite sum if and only if the absolute value of its common ratio 'r' is less than 1 (i.e., |r| < 1).
What happens if |r| >= 1 for an infinite geometric series?: If |r| >= 1 and a ≠ 0, the infinite geometric series diverges, meaning its sum is not a finite number (it goes to infinity or oscillates without approaching a limit).
What if the common ratio r = 1 in a finite series?: If r = 1, all terms are the same (a), and the sum of n terms is simply n * a. The standard formula has a (1-r) denominator, so it's undefined, but the sum is clear.
Can the common ratio be negative?: Yes, the common ratio 'r' can be negative. This results in terms alternating in sign (e.g., 2, -4, 8, -16…).
How is the Sum of Geometric Series Calculator useful in finance?: It can model things like the present value of an annuity or perpetuity, or the future value of investments with growth rates, where payments or values grow at a constant ratio.
Can I use this calculator for a decreasing series?: Yes, if the common ratio 'r' is between 0 and 1 (or -1 and 0), the absolute values of the terms decrease.
What if my first term is 0?: If the first term 'a' is 0, all terms are 0, and the sum of the series is 0, regardless of 'r' or 'n'.

Related Tools and Internal Resources

Arithmetic Series Calculator: Calculate the sum of an arithmetic series.
Compound Interest Calculator: See how investments grow with compounding, which relates to geometric series.
Annuity Calculator: Calculate payments or present/future values of annuities, often involving geometric series concepts.
Present Value Calculator: Find the present value of future cash flows.
Future Value Calculator: Project the future value of investments.
Sequence and Series Explained: Learn more about different types of sequences and series.

Find The Sum Of The Geometric Series Calculator