Geometric Progression Calculator

Calculate the terms and sum of a geometric sequence instantly.

Progression Visualization

First 10 terms of the geometric progression.

Term (n)	Value (a_n)

What is a Geometric Progression Calculator?

A geometric progression calculator is a tool designed to analyze 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. This calculator helps you compute key properties of a geometric sequence, such as the value of a specific term (the nth term), the sum of a finite number of terms, and the sum of an infinite series if it converges.

This tool is invaluable for students, financiers, engineers, and anyone dealing with exponential growth or decay scenarios. Whether you are studying for a math exam, calculating compound interest, or modeling population growth, a reliable geometric progression calculator simplifies complex calculations.

Geometric Progression Formula and Explanation

The core of a geometric progression (GP) is defined by its first term, common ratio, and the number of terms. The primary formulas used by this geometric progression calculator are:

Nth Term Formula

To find the nth term (a_n) of a GP, you use the formula:

a_n = a * r^(n-1)

Sum of the First n Terms (S_n)

The sum of the first n terms is calculated differently based on the common ratio (r):

• If r ≠ 1: S_n = a * (1 - r^n) / (1 - r)
• If r = 1: S_n = a * n

Sum to Infinity (S_∞)

An infinite geometric series has a finite sum only if the absolute value of the common ratio is less than 1 (i.e., |r| < 1). The formula is:

S_∞ = a / (1 - r)

Variables Table

Variable	Meaning	Unit	Typical Range
a	First Term	Unitless or context-dependent (e.g., currency, population)	Any real number
r	Common Ratio	Unitless	Any real number
n	Number of Terms	Unitless	Positive integer (>0)

Practical Examples

Example 1: Compound Interest

Imagine you invest $1,000 in an account with a 5% annual compound interest rate. This is a real-world geometric progression.

• Inputs:
  • First Term (a): $1,000
  • Common Ratio (r): 1.05 (since it’s a 5% increase)
  • Number of Terms (n): 10 (for 10 years)
• Results:
  • Value after 10 years (the 10th term): a * r^(10-1) = $1,000 * 1.05^9 ≈ $1,551.33. The total amount after 10 years is actually the 11th term in the sequence (start, year 1, year 2…), which is $1,000 * 1.05^10 ≈ $1,628.89.

Example 2: Population Decline

A city’s population of 50,000 is decreasing by 2% each year.

• Inputs:
  • First Term (a): 50,000
  • Common Ratio (r): 0.98 (a 2% decrease)
  • Number of Terms (n): 5 (to find the population in 5 years)
• Results:
  • Population in 5 years (the 6th term): 50,000 * 0.98^5 ≈ 45,196.

How to Use This Geometric Progression Calculator

Using the calculator is straightforward. Follow these steps for an accurate calculation:

1. Enter the First Term (a): Input the initial value of your sequence.
2. Enter the Common Ratio (r): Input the multiplier for the sequence. For a 5% increase, `r` would be 1.05. For a 5% decrease, `r` would be 0.95.
3. Enter the Number of Terms (n): Input how many terms you want to analyze or sum. This must be a positive integer.
4. Interpret the Results: The calculator automatically updates the nth term, the sum of n terms, and other relevant values as you type. The results are clearly labeled for you to use. The chart and table below the calculator also visualize the progression.

For more related tools, check out our arithmetic progression calculator.

Key Factors That Affect Geometric Progression

• The Sign of the First Term (a): A positive ‘a’ means all terms will have the same sign as ‘r’ dictates (if r is positive). A negative ‘a’ will invert the signs.
• The Magnitude of the Common Ratio (|r|): This is the most critical factor. If |r| > 1, the sequence grows exponentially (diverges). If |r| < 1, the sequence shrinks towards zero (converges). If |r| = 1, the sequence is constant.
• The Sign of the Common Ratio (r): A positive ‘r’ results in a sequence where all terms have the same sign. A negative ‘r’ results in an alternating sequence (e.g., 5, -10, 20, -40).
• The Number of Terms (n): For a divergent series, a larger ‘n’ leads to extremely large term values and sums. For a convergent series, a larger ‘n’ brings the sum closer to the sum to infinity.
• Integer vs. Fractional Ratios: Integer ratios lead to rapid growth, while fractional ratios (between -1 and 1) lead to decay.
• Initial Value (a): The starting term ‘a’ scales the entire sequence. A larger ‘a’ will result in larger values for all subsequent terms, but it does not affect whether the series converges or diverges. For more information on sequences, see our guide on sequence and series.

FAQ

What is the difference between a geometric and an arithmetic progression?
An arithmetic progression involves adding a constant difference to get the next term, while a geometric progression involves multiplying by a constant ratio.

How do I find the common ratio?
Divide any term by its preceding term. For example, in the sequence 2, 4, 8, 16, the common ratio is 4/2 = 2. You can learn more with an nth term calculator.

Can the common ratio be negative?
Yes. A negative common ratio means the terms will alternate in sign (e.g., positive, negative, positive, etc.).

When does a geometric series have a finite sum to infinity?
A geometric series can be summed to infinity only if the absolute value of its common ratio `r` is less than 1 (i.e., -1 < r < 1). If |r| ≥ 1, the sum is infinite (it diverges).

What happens if the common ratio (r) is 1?
If r = 1, the sequence is a constant sequence (e.g., 5, 5, 5, …). The sum of n terms is simply n * a. Our geometric progression calculator handles this edge case correctly.

What happens if the common ratio (r) is 0?
If r = 0, all terms after the first term will be zero (e.g., 5, 0, 0, 0, …).

How is the common ratio formula applied in real life?
The common ratio formula is key in calculating compound interest, population growth rates, radioactive decay, and viral spread.

Can I calculate the sum of an infinite geometric series?
Yes, but only if it converges (|r| < 1). Our calculator will provide the "Sum to Infinity" if this condition is met; otherwise, it will indicate that the sum is infinite. See our tool for infinite geometric series.

Related Tools and Internal Resources

Explore other useful calculators and resources to deepen your understanding of mathematical sequences and financial calculations.

• Arithmetic Progression Calculator: For sequences with a common difference.
• Compound Interest Calculator: A practical application of geometric progressions in finance.
• General Math Calculators: A collection of various mathematical tools.