Find The Sum Of Geometric Series Calculator

Calculate the sum of the first 'n' terms of a geometric series using this sum of geometric series calculator.

Term Details & Visualization

Term (i)	Term Value (a*r^(i-1))	Cumulative Sum
Enter values and calculate to see details.

Table showing the first few term values and their cumulative sum.

What is a Sum of Geometric Series Calculator?

A sum of geometric series calculator is a tool used to find the total sum of a finite number of terms in a geometric sequence (also known as a geometric progression). In a geometric sequence, each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This calculator automates the process of applying the sum formula, making it easy to find the sum without manual calculations, especially for a large number of terms.

This calculator is useful for students studying sequences and series in mathematics, finance professionals analyzing investments with compounding growth, engineers, and anyone dealing with processes that exhibit exponential growth or decay. The sum of geometric series calculator helps in understanding the cumulative effect over several periods.

Common misconceptions include confusing it with an arithmetic series (where terms are added by a constant difference) or thinking it only applies to increasing series (it works for decreasing series too, where the common ratio is between -1 and 1, excluding 0).

Sum of Geometric Series Formula and Mathematical Explanation

A geometric series is a series whose terms form a geometric sequence. The sum of the first 'n' terms of a geometric series is denoted by S_n.

Let the geometric sequence be: a, ar, ar², ar³, …, ar^n-1

Where:

• 'a' is the first term
• 'r' is the common ratio
• 'n' is the number of terms

The sum S_n is given by: S_n = a + ar + ar² + … + ar^n-1

To derive the formula, multiply S_n by 'r': rS_n = ar + ar² + ar³ + … + arⁿ

Subtract the second equation from the first:

S_n – rS_n = (a + ar + … + ar^n-1) – (ar + ar² + … + arⁿ)

S_n(1 – r) = a – arⁿ

S_n(1 – r) = a(1 – rⁿ)

If r ≠ 1, we can divide by (1 – r):

S_n = a(1 – rⁿ) / (1 – r)

If r = 1, the series is a, a, a, …, a, and the sum is simply:

S_n = n × a

Our sum of geometric series calculator uses these formulas based on the value of '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, …)
Sn	Sum of the first 'n' terms	Same as 'a'	Depends on a, r, n

Practical Examples (Real-World Use Cases)

Example 1: Savings Growth

Suppose you save $100 in the first month, and each month you save 5% more than the previous month. How much will you have saved after 12 months?

• First term (a) = 100
• Common ratio (r) = 1 + 0.05 = 1.05
• Number of terms (n) = 12

Using the sum of geometric series calculator with these inputs: S₁₂ = 100(1 – 1.05¹²) / (1 – 1.05) ≈ 100(1 – 1.795856) / (-0.05) ≈ 100(-0.795856) / (-0.05) ≈ $1591.71. You would have saved approximately $1591.71 after 12 months.

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. What is the total distance traveled by the ball downwards before it practically stops (say, after 10 bounces, considering only downward distance for each bounce after the first drop)?

• First term (a) = 10 (initial drop)
• Common ratio (r) = 0.70
• Number of terms (n) = 10

Using the sum of geometric series calculator: S₁₀ = 10(1 – 0.70¹⁰) / (1 – 0.70) ≈ 10(1 – 0.0282475) / 0.30 ≈ 10(0.9717525) / 0.30 ≈ 32.39 meters. The total downward distance after 10 bounces is about 32.39 meters. (Note: Total distance would include upward travel too).

How to Use This Sum of Geometric Series Calculator

1. Enter the First Term (a): Input the initial value of your geometric sequence.
2. Enter the Common Ratio (r): Input the factor by which each term is multiplied to get the next term.
3. Enter the Number of Terms (n): Input the total number of terms you wish to sum. This must be a positive integer.
4. Calculate: The calculator will automatically display the sum (S_n) and intermediate values as you input or change the numbers. You can also click the "Calculate Sum" button.
5. Read the Results: The primary result is the sum S_n. You also see the inputs a, r, and n used for the calculation.
6. View Details: The table and chart show the values of the first few terms and their cumulative sum, providing a visual representation if 'n' is not too large.

Use the "Reset" button to clear inputs and "Copy Results" to copy the main findings.

Key Factors That Affect Sum of Geometric Series Results

The sum of a geometric series is primarily influenced by:

1. First Term (a): The larger the initial term, the larger the sum, assuming other factors are constant and positive.
2. Common Ratio (r): This is a crucial factor.
   • If |r| > 1, the terms grow in magnitude, and the sum can become very large (or very negative) quickly as 'n' increases.
   • If |r| < 1, the terms decrease in magnitude, and the sum approaches a finite limit as 'n' goes to infinity (see our infinite geometric series sum calculator).
   • If r = 1, the sum is simply n*a.
   • If r is negative, the terms alternate in sign.
3. Number of Terms (n): The more terms you sum, the larger the magnitude of the sum will generally be, especially if |r| > 1. For |r| < 1, the sum will converge as n increases.
4. Sign of 'a' and 'r': The signs of 'a' and 'r' determine the sign of individual terms and thus the overall sum.
5. Magnitude of 'r' relative to 1: Whether |r| is greater than, less than, or equal to 1 dramatically changes the behavior of the sum as 'n' increases.
6. Integer value of 'n': 'n' must be a positive integer representing the count of terms.

Understanding these factors helps in predicting how the sum will behave. The sum of geometric series calculator instantly shows these effects.

Frequently Asked Questions (FAQ)

What is a geometric series?

A geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

What's the difference between a geometric sequence and a geometric series?

A geometric sequence is a list of numbers (e.g., 2, 4, 8, 16,…), while a geometric series is the sum of those numbers (e.g., 2 + 4 + 8 + 16 + …).

Can the common ratio (r) be negative?

Yes, if 'r' is negative, the terms of the series will alternate in sign (e.g., 10, -5, 2.5, -1.25,…).

What happens if the common ratio (r) is 1?

If r = 1, all terms are the same (a, a, a,…), and the sum is simply n * a. Our sum of geometric series calculator handles this case.

What happens if the common ratio (r) is 0?

If r = 0, all terms after the first are zero (a, 0, 0,…), and the sum is just 'a' for n >= 1.

Can I use the sum of geometric series calculator for an infinite series?

This calculator is for a *finite* number of terms ('n'). For an infinite series, the sum converges only if |r| < 1, and the sum is a / (1 - r). We have a separate infinite geometric series sum calculator for that.

How do I find the common ratio?

If you have two consecutive terms, divide the second term by the first term. You can use a common ratio calculator if needed.

What if I need to find a specific term, not the sum?

You can use the formula a_n = ar^n-1 or our nth term of geometric sequence calculator.

Related Tools and Internal Resources

• Geometric Progression Calculator: Explore properties of geometric sequences.
• Finite Series Calculator: General tools for finite series sums.
• Infinite Geometric Series Sum: Calculate the sum when the number of terms is infinite and |r| < 1.
• Common Ratio Calculator: Find the common ratio of a geometric sequence.
• Nth Term of Geometric Sequence: Find a specific term in a geometric sequence.
• Sequence Calculator: General tools for various number sequences.