1. What is Common Ratio?

The common ratio (r) is a fundamental concept in geometric sequences, representing the constant factor between consecutive terms. It is calculated by dividing any term by its preceding term in the sequence.

2. How Does the Calculator Work?

The calculator uses the common ratio formula:

Where:

• \( term_{n+1} \) — The (n+1)th term in the sequence
• \( term_n \) — The nth term in the sequence
• \( r \) — Common ratio (unitless)

Explanation: The formula calculates the constant multiplier that relates consecutive terms in a geometric sequence.

3. Importance of Common Ratio

Details: The common ratio is essential for identifying geometric sequences, predicting future terms, calculating sums of geometric series, and solving various mathematical and real-world problems involving exponential growth or decay.

4. Using the Calculator

Tips: Enter any two consecutive terms from a geometric sequence. The denominator (term n) cannot be zero. The result is unitless as it represents a ratio.

5. Frequently Asked Questions (FAQ)

Q1: What defines a geometric sequence?
A: A geometric sequence is one 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, the common ratio can be positive, negative, or even a fraction. A negative ratio creates an alternating sequence.

Q3: What if the common ratio is zero?
A: If the common ratio is zero, all subsequent terms after the first non-zero term will be zero, creating a truncated 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 practical applications of common ratio?
A: Common ratio is used in finance (compound interest), physics (exponential decay), computer science (algorithm analysis), and population growth models.