Common Ratio of Geometric Progression given Last Term Calculator

Common Ratio Formula:

\[ r = \left( \frac{l}{a} \right)^{\frac{1}{n_{Total}-1}} \]

Common Ratio of Progression (r):

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1. What is the Common Ratio of Geometric Progression?

The Common Ratio of a Geometric Progression is the constant factor between consecutive terms in the sequence. It determines the pattern of growth or decay in the progression and is a fundamental property of geometric sequences.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r = \left( \frac{l}{a} \right)^{\frac{1}{n-1}} \]

Where:

\( r \) — Common Ratio of Progression
\( l \) — Last Term of Progression
\( a \) — First Term of Progression
\( n \) — Number of Total Terms of Progression

Explanation: The formula calculates the common ratio by finding the (n-1)th root of the ratio between the last and first terms of the geometric progression.

3. Importance of Common Ratio Calculation

Details: Calculating the common ratio is essential for understanding the behavior of geometric sequences, predicting future terms, and solving problems in mathematics, finance, and various scientific fields.

4. Using the Calculator

Tips: Enter the first term, last term, and total number of terms in the progression. All values must be positive numbers, and the total number of terms must be at least 2.

5. Frequently Asked Questions (FAQ)

Q1: What is a geometric progression?
A: A geometric progression 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.

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 happens if the common ratio is between 0 and 1?
A: If 0 < r < 1, the geometric progression shows exponential decay, with each term getting smaller than the previous one.

Q4: What if the common ratio is greater than 1?
A: If r > 1, the geometric progression shows exponential growth, with each term getting larger than the previous one.

Q5: Can the common ratio be zero?
A: No, the common ratio cannot be zero in a geometric progression, as it would make all terms after the first term equal to zero.