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Common Ratio Formula:
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The common ratio (r) in a geometric sequence is the constant factor between consecutive terms. It determines how the sequence grows or decays and is fundamental to understanding geometric progressions.
The calculator uses the common ratio formula:
Where:
Explanation: The common ratio is calculated by dividing any term in the sequence by the previous term. This ratio remains constant throughout a geometric sequence.
Details: Calculating the common ratio is essential for understanding the behavior of geometric sequences, predicting future terms, and solving problems in mathematics, finance, and physics that involve exponential growth or decay.
Tips: Enter the next term and current term values in the same units. Both values must be positive numbers greater than zero for accurate calculation.
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 negative, which results in an alternating sequence where terms switch between positive and negative values.
Q3: What does a common ratio greater than 1 indicate?
A: A common ratio greater than 1 indicates exponential growth in the sequence, where each term is larger than the previous one.
Q4: What does a common ratio between 0 and 1 indicate?
A: A common ratio between 0 and 1 indicates exponential decay, where each term is smaller than the previous one but still positive.
Q5: How is the common ratio used in real-world applications?
A: The common ratio is used in calculating compound interest, population growth, radioactive decay, and in various financial and scientific models involving exponential change.