Series Parameters
Sequence Results
A Geometric Series Calculator is a powerful mathematical tool designed to analyze and calculate the properties of a geometric progression. It instantly finds the value of any specific term in the sequence, the total sum of multiple terms, and determines if the sequence has a finite sum to infinity. This is widely used in compound interest calculations, population growth modeling, and fractal geometry.
Understanding Geometric Sequences
A geometric sequence is a list of numbers where every term after the first is found by multiplying the previous term by a fixed, non-zero number. This constant multiplier is known as the "Common Ratio".
To find the N-th Term in a sequence, you use the formula:
aₙ = a × r^(n-1)
Where a is the first term, r is the common ratio, and n is the position of the term you want to find.
How to Calculate the Sum (Sn)
The sum of a geometric series is the total result when you add all the terms together. The formula changes slightly depending on the common ratio.
If the ratio is not exactly 1, the formula is: Sₙ = a × (1 - r^n) / (1 - r)
If the ratio is a fraction between -1 and 1, the numbers get smaller and smaller as the sequence goes on. Because of this shrinking nature, you can actually calculate the total sum of an infinite number of terms using the Sum to Infinity formula: S∞ = a / (1 - r).
How to Use This Calculator
- Enter the First Term (a) of your sequence. This is your starting number.
- Enter the Common Ratio (r). If your sequence doubles every time, enter 2. If it gets cut in half, enter 0.5.
- Enter the Number of Terms (n) you want to analyze.
- The calculator instantly provides the value of the N-th term, the sum of all N terms, and the sum to infinity if it exists.
Frequently Asked Questions
What does "Divergent" mean in the sum to infinity?
A series is divergent if its numbers keep getting larger and larger infinitely (which happens when the common ratio is greater than 1 or less than -1). Because the numbers never stop growing, it is impossible to calculate a final infinite sum, so the answer is strictly divergent.
Can the common ratio be negative?
Yes. If the common ratio is a negative number, your sequence will become "alternating." This means the numbers will flip back and forth between positive and negative values. For example, starting with 2 and a ratio of -3 gives you 2, -6, 18, -54.
What is the difference between geometric and arithmetic series?
In a geometric series, you MULTIPLY the same number to get the next term. In an arithmetic series, you ADD the same number to get the next term.