What is a Geometric Sequence?
A geometric sequence, also known as a geometric progression, is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding geometric sequences is essential in fields ranging from finance (compound interest) to biology (population growth) and computer science.
How to Use This Calculator
Using our Geometric Sequence Calculator with steps is straightforward. Simply input the following three values:
- First Term (a₁): The starting value of your sequence.
- Common Ratio (r): The factor by which you multiply each term to get the next.
- Number of Terms (n): How many terms you want to calculate or find the sum for.
Once you click calculate, the tool will provide you with the nth term value, the sum of the series, and a detailed step-by-step breakdown of how each term was derived.
Geometric Sequence Formulas
The calculator uses two primary formulas to provide results:
- The Nth Term Formula: aₙ = a₁ × r^(n-1)
- The Sum Formula: Sₙ = a₁(1 - rⁿ) / (1 - r), provided that r ≠ 1. If r = 1, the sum is simply a₁ × n.
Frequently Asked Questions
What is the common ratio?
The common ratio is the constant value used to multiply one term to reach the next. You can find it by dividing any term by its preceding term (e.g., a₂ / a₁).
Can the common ratio be negative?
Yes, if the common ratio is negative, the terms in the sequence will alternate between positive and negative values.
What happens if the ratio is 1?
If the common ratio is 1, every term in the sequence is identical to the first term, making it both a geometric and an arithmetic sequence.