Geometric Sequence Ratio Calculator

Geometric Sequence Ratio Formula:

\[ r = \frac{a_{k+1}}{a_k} \]

Common Ratio (r):

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1. What is the Geometric Sequence Ratio?

The common ratio (r) in a geometric sequence is the constant factor between consecutive terms. It determines how each term relates to the previous one and defines the pattern of growth or decay in the sequence.

2. How Does the Calculator Work?

The calculator uses the geometric sequence ratio formula:

\[ r = \frac{a_{k+1}}{a_k} \]

Where:

\( a_{k+1} \) — Next term in the sequence (unit of sequence)
\( a_k \) — Current term in the sequence (unit of sequence)
\( r \) — Common ratio (unitless)

Explanation: The common ratio is calculated by dividing any term by its preceding term in a geometric sequence.

3. Importance of Common Ratio Calculation

Details: Calculating the common ratio is essential for understanding geometric sequences, predicting future terms, analyzing exponential growth or decay patterns, and solving problems in mathematics, finance, and science.

4. Using the Calculator

Tips: Enter the current term and next term values. Both values must be valid numbers, and the current term cannot be zero (division by zero is undefined).

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, the common ratio can be positive, negative, or even a fraction. A negative ratio causes the sequence to alternate between positive and negative values.

Q3: What if the common ratio is between 0 and 1?
A: If 0 < r < 1, the sequence shows exponential decay. If r > 1, the sequence shows exponential growth.

Q4: Can the common ratio be zero?
A: No, the common ratio cannot be zero because it would make all subsequent terms zero, breaking the geometric sequence pattern.

Q5: How is the common ratio used in real-world applications?
A: The common ratio is used in compound interest calculations, population growth models, radioactive decay, and many other exponential growth/decay scenarios.