Find The Sum Of A Geometric Series Calculator

Term (k)	Value (a_k)	Partial Sum (S_k)
Enter values and calculate to see the table.

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 (also known as a geometric progression). A geometric sequence 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). If the first term is 'a' and the common ratio is 'r', the sequence looks like: a, ar, ar², ar³, …

Our find the sum of a geometric series calculator helps you compute this sum quickly for a finite number of terms (S_n) and also for an infinite series (S_∞) if it converges (|r| < 1).

Who Should Use This Calculator?

This calculator is useful for students studying sequences and series in mathematics, finance professionals dealing with annuities or compound interest scenarios that follow a geometric progression, engineers, and anyone needing to sum a series of numbers that grow or decrease by a constant ratio.

Common Misconceptions

A common misconception is that all geometric series have a finite sum. This is only true for infinite series if the absolute value of the common ratio is less than 1 (|r| < 1). Otherwise, the sum of an infinite geometric series diverges (goes to infinity or negative infinity, or oscillates). For a finite number of terms, the sum is always finite. Another point is not to confuse it with an arithmetic series, where terms have a common *difference*, not a ratio.

Sum of a Geometric Series Formula and Mathematical Explanation

The sum of the first 'n' terms of a geometric series (S_n) is given by the formula:

If r ≠ 1: S_n = a(1 – rⁿ) / (1 – r)

If r = 1: S_n = n * a (as all terms are 'a')

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

The k-th term (a_k) is given by a_k = a * r^(k-1).

If the absolute value of the common ratio is less than 1 (|r| < 1), the sum of an infinite geometric series (S_∞) converges to:

S_∞ = a / (1 – r)

Variables Table

Variable	Meaning	Unit	Typical Range
a	First term	Unitless or units of the quantity	Any real number
r	Common ratio	Unitless	Any real number
n	Number of terms	Unitless	Positive integer (1, 2, 3, …)
S_n	Sum of the first n terms	Same as 'a'	Depends on a, r, n
a_n	The n-th term	Same as 'a'	Depends on a, r, n
S_∞	Sum to infinity	Same as 'a'	Finite if \|r\| < 1, otherwise diverges

Practical Examples (Real-World Use Cases)

Example 1: Savings Growth

Imagine you save $100 in the first month, and each month you save 10% more than the previous month. How much will you have saved after 6 months?

First term (a) = 100
Common ratio (r) = 1 + 0.10 = 1.1
Number of terms (n) = 6

Using the formula S_n = 100(1 – 1.1⁶) / (1 – 1.1) = 100(1 – 1.771561) / (-0.1) = 100(-0.771561) / (-0.1) = 771.561. You would have saved approximately $771.56 after 6 months. Our find the sum of a geometric series calculator can compute this instantly.

Example 2: Bouncing Ball

A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 70% of the previous height. What is the total vertical distance traveled by the ball before it comes to rest?

The distances are: 10 (down), 10*0.7 (up), 10*0.7 (down), 10*0.7*0.7 (up), 10*0.7*0.7 (down), and so on.

Initial drop: 10m.

Subsequent bounces (up and down): 2 * (10*0.7 + 10*0.7² + 10*0.7³ + …). This is an infinite geometric series with a = 10*0.7 = 7, r = 0.7.

Sum to infinity of bounces = 7 / (1 – 0.7) = 7 / 0.3 = 70/3 = 23.33 m.

Total distance = 10 (initial drop) + 2 * 23.33 = 10 + 46.66 = 56.66 meters. The geometric progression calculator can help analyze the heights.

How to Use This Find the Sum of a Geometric Series Calculator

Enter the First Term (a): Input the initial value of your series.
Enter the Common Ratio (r): Input the factor by which each term is multiplied to get the next.
Enter the Number of Terms (n): For a finite series, enter the total number of terms you want to sum. This must be a positive integer.
View Results: The calculator automatically updates the Sum (S_n), Last Term (a_n), and Sum to Infinity (S_∞, if |r| < 1). The table and chart also update.
Read Explanation: The formulas used are displayed for clarity.

The results allow you to quickly understand the total value accumulated over 'n' terms, the value of the last term, and the limiting sum if the series converges. For more complex series, a series calculator might be useful.

Key Factors That Affect Sum of a Geometric Series Results

First Term (a): The starting value directly scales the sum. A larger 'a' leads to a proportionally larger sum.
Common Ratio (r): This is the most critical factor.
- If |r| > 1, the terms grow exponentially, and the sum increases rapidly with 'n'.
- If |r| < 1, the terms decrease, and the sum approaches a finite limit as 'n' increases (S_∞).
- If r = 1, the sum is simply n*a.
- If r is negative, the terms alternate in sign.
Number of Terms (n): For |r| > 1, increasing 'n' drastically increases the sum. For |r| < 1, increasing 'n' brings the sum closer to S_∞.
Sign of 'a' and 'r': The signs of 'a' and 'r' determine the sign of the terms and thus the sum. If 'r' is negative, terms alternate.
Proximity of |r| to 1: When |r| is close to 1 (but not 1), the sum can be very sensitive to 'n', especially for large 'n'.
Convergence Condition (|r| < 1): Whether an infinite series has a finite sum depends solely on |r| being less than 1. You can explore understanding sequences to learn more.

Frequently Asked Questions (FAQ)

Q: What happens if the common ratio (r) is 1? A: If r=1, each term is the same as the first term 'a'. The sum of 'n' terms is simply n * a. Our find the sum of a geometric series calculator handles this case.

Q: What if the common ratio (r) is negative? A: The terms will alternate in sign. The formulas still apply. If |r| < 1, the sum to infinity still converges.

Q: Can the number of terms (n) be zero or negative? A: No, 'n' must be a positive integer representing the number of terms you are summing.

Q: When does an infinite geometric series have a finite sum? A: An infinite geometric series has a finite sum (converges) only when the absolute value of the common ratio is less than 1 (i.e., -1 < r < 1). The calculator shows S_∞ in this case. Check our math solver for other problems.

Q: What is the difference between a geometric and arithmetic series? A: In a geometric series, each term is multiplied by a common ratio to get the next term. In an arithmetic series, a common difference is added to each term to get the next. We have an arithmetic series calculator too.

Q: How do I find the common ratio? A: Divide any term by its preceding term. For example, the second term divided by the first, or the third by the second.

Q: What if |r| > 1 and I want the sum to infinity? A: The sum to infinity diverges (goes to infinity or negative infinity) and does not have a finite value.

Q: Can 'a' be zero? A: Yes, if 'a' is zero, all terms are zero, and the sum is zero.

Related Tools and Internal Resources

Arithmetic Series Calculator: Calculate the sum of an arithmetic series.
Compound Interest Calculator: Compound interest often follows a geometric progression.
Geometric Sequence Calculator: Focuses on individual terms of a geometric sequence.
Series Calculator: A more general tool for various mathematical series.
Math Solver: Solve a variety of math problems.
Understanding Sequences and Series: A guide to the concepts of mathematical sequences.

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