1. What is Common Ratio in Geometric Sequence?

The common ratio (r) in a geometric sequence is the constant factor between consecutive terms. It is found by dividing any term by its preceding term. A geometric sequence follows the pattern: a, ar, ar², ar³, ..., where 'a' is the first term and 'r' is the common ratio.

2. How Does the Calculator Work?

The calculator uses the common ratio formula:

Where:

• \( a_n \) — The nth term in the sequence
• \( a_{n-1} \) — The term immediately preceding the nth term
• \( r \) — The common ratio

Explanation: This formula calculates the constant factor that multiplies each term to get the next term in a geometric sequence.

3. Importance of Common Ratio

Details: The common ratio is fundamental to understanding geometric sequences and series. It determines whether the sequence is increasing, decreasing, alternating, or constant. It's essential for calculating future terms, sums of sequences, and analyzing exponential growth or decay patterns.

4. Using the Calculator

Tips: Enter any two consecutive terms from a geometric sequence. The calculator will compute their ratio. Ensure the denominator (a_{n-1}) is not zero to avoid division errors.

5. Frequently Asked Questions (FAQ)

Q1: What if the common ratio is negative?
A: A negative common ratio creates an alternating sequence where terms switch between positive and negative values.

Q2: What if the common ratio is between 0 and 1?
A: A common ratio between 0 and 1 creates a decreasing geometric sequence that approaches zero.

Q3: What if the common ratio is exactly 1?
A: A common ratio of 1 creates a constant sequence where all terms are equal.

Q4: Can the common ratio be zero?
A: If the common ratio is zero, all terms after the first will be zero, creating a truncated sequence.

Q5: 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).