1. What is the Common Ratio?

The common ratio is a fundamental concept in geometric sequences, representing the constant factor between consecutive terms. It determines the growth or decay pattern of the sequence.

2. How Does the Calculator Work?

The calculator uses the common ratio formula:

Where:

• \( r \) — Common ratio (dimensionless)
• \( a_{n+1} \) — Next term in the sequence
• \( a_n \) — Current term in the sequence

Explanation: The common ratio is calculated by dividing any term in the geometric sequence by its preceding term. This ratio remains constant throughout the sequence.

3. Importance of Common Ratio

Details: The common ratio is crucial for understanding geometric sequences and series. It determines whether the sequence grows (r > 1), decays (0 < r < 1), alternates (r < 0), or remains constant (r = 1).

4. Using the Calculator

Tips: Enter the next term and current term values. Both values must be positive and non-zero. The calculator will compute the common ratio, which is dimensionless.

5. Frequently Asked Questions (FAQ)

Q1: What is a geometric sequence?
A: A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q2: Can the common ratio be negative?
A: Yes, a negative common ratio creates an alternating sequence where terms switch between positive and negative values.

Q3: What happens if the common ratio is zero?
A: If r = 0, all terms after the first become zero, creating a degenerate geometric sequence.

Q4: How is common ratio different from common difference?
A: Common ratio involves multiplication between terms (geometric sequence), while common difference involves addition between terms (arithmetic sequence).

Q5: What are real-world applications of geometric sequences?
A: Geometric sequences model exponential growth/decay in populations, investments, radioactive decay, and many natural phenomena.