Finding the Common Ratio in a Geometric Progression
In a geometric progression, the terms follow a specific pattern where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This article will guide you through the process of finding the common ratio when the first and third terms of the geometric progression are given as 3/4 and 3/25 respectively.
Understanding the Concept of Geometric Progression
A geometric progression (GP) is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. The general form of a GP can be expressed as:
$T_n a cdot r^{(n-1)}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.Solving for the Common Ratio
Given:
The first term, $T_1 a frac{3}{4}$ The third term, $T_3 a cdot r^2 frac{3}{25}$We can set up the equation:
$frac{3}{4}r^2 frac{3}{25}$
Divide both sides by $frac{3}{4}$
$r^2 frac{3}{25} div frac{3}{4} frac{3}{25} times frac{4}{3} frac{4}{25}$
Take the square root of both sides:
$r sqrt{frac{4}{25}} frac{2}{5}$
The common ratio, $r$, is $frac{2}{5}$.
Verifying the Solution
To ensure our solution is correct, we can check if the terms of the geometric progression are consistent:
The first term, $T_1 frac{3}{4}$. The second term, $T_2 frac{3}{4} cdot frac{2}{5} frac{6}{20} frac{3}{10}$. The third term, $T_3 frac{3}{10} cdot frac{2}{5} frac{6}{50} frac{3}{25}$, which matches the given value.This confirms that the common ratio $r frac{2}{5}$ is correct.
Conclusion
In this article, we have demonstrated how to find the common ratio in a geometric progression given two terms. The process involves setting up and solving a simple algebraic equation. The common ratio for the given terms is $frac{2}{5}$. Understanding this concept and its application can be useful in various fields, including mathematics, science, and engineering.
Keywords: geometric progression, common ratio, algebraic equations