1. What is a Geometric Sequence?

A geometric sequence is a sequence 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. Unlike arithmetic sequences with a common difference, geometric sequences multiply by a constant factor.

2. How Does the Calculator Work?

The calculator uses the geometric sequence formula:

Where:

• \( a_1 \) — First term of the sequence
• \( a_n \) — nth term of the sequence
• \( r \) — Common ratio
• \( n \) — Term position

Explanation: To find the common ratio between two consecutive terms, simply divide the second term by the first term: \( r = \frac{a_2}{a_1} \)

3. Importance of Common Ratio

Details: The common ratio determines the growth or decay pattern of the sequence. A ratio greater than 1 indicates exponential growth, while a ratio between 0 and 1 indicates exponential decay. Negative ratios create alternating sequences.

4. Using the Calculator

Tips: Enter the first and second terms of your geometric sequence. The calculator will compute the common ratio. Ensure the first term is not zero to avoid division by zero errors.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between arithmetic and geometric sequences?
A: Arithmetic sequences add a constant difference, while geometric sequences multiply by a constant 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 if the common ratio is zero?
A: If the common ratio is zero, all terms after the first will be zero, creating a trivial geometric sequence.

Q4: How do I find subsequent terms?
A: Multiply any term by the common ratio to get the next term in the sequence.

Q5: What are real-world applications of geometric sequences?
A: Geometric sequences model population growth, radioactive decay, compound interest, and many natural phenomena that follow exponential patterns.