1. What is Common Ratio?

The common ratio (r) 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 formula calculates the ratio between any two consecutive terms in a geometric sequence, which remains constant throughout the sequence.

3. Importance of Common Ratio

Details: The common ratio is essential for understanding geometric sequences, predicting future terms, calculating sums of geometric series, and modeling exponential growth or decay in real-world applications.

4. Using the Calculator

Tips: Enter the value of the next term (a_{n+1}) and the current term (a_n). The current term must not be zero. The result is dimensionless and represents the multiplicative factor between terms.

5. Frequently Asked Questions (FAQ)

Q1: What does the common ratio tell us about a sequence?
A: The common ratio indicates whether the sequence is increasing (|r| > 1), decreasing (0 < |r| < 1), alternating (r < 0), or constant (r = 1).

Q2: Can the common ratio be zero?
A: Yes, but if r = 0, all subsequent terms after the first non-zero term will be zero, creating a truncated sequence.

Q3: What if the common ratio is negative?
A: A negative common ratio creates an alternating sequence where terms switch between positive and negative values.

Q4: How is common ratio different from common difference?
A: Common ratio is for geometric sequences (multiplicative), while common difference is for arithmetic sequences (additive).

Q5: What are practical applications of common ratio?
A: Used in finance (compound interest), physics (radioactive decay), computer science (algorithm analysis), and population growth models.