Understanding the common ratio of a geometric progression is fundamental in mathematics. This article expounds the process of finding the common ratio when the 6th term and the first term of the progression are given. The key case study involves a geometric progression where the 6th term is 2/27 and the first term is 18. We will demonstrate the step-by-step procedure to determine the common ratio, along with the mathematical reasoning behind each step.

Introduction to Geometric Progression

A geometric progression (GP) is a sequence of numbers where each successive term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula for the nth term of a GP is given by:

an a1rn-1

Where:

a1 is the first term, r is the common ratio, and n is the term number.

The Problem: Finding the Common Ratio

In this particular problem, we are given that the 6th term (a6) of a geometric progression is 2/27 and the first term (a1) is 18. We need to find the common ratio (r).

Step 1: Using the Formula for the 6th Term

The 6th term formula is:

a6 a1r5

Substituting the given values:

2/27 18r5

Step 2: Isolating the Term Involving the Ratio

To isolate r5, we divide both sides by 18:

2/27 / 18 r5

This simplifies to:

1/243 r5

Step 3: Finding the 5th Root

Since r5 1/243, we find the 5th root of r5:

r i?rt(1/243) 1/3

Therefore, the common ratio (r) is 1/3.

Verification and Mathematical Insight

To verify, we can substitute r 1/3 back into the formula for the 6th term:

a6 18(1/3)5 18(1/243) 2/27

Since the original 6th term is 2/27, this confirms our solution.

Further Exploration: The Geometric Sequence

Given the common ratio r 1/3, we can generate the first few terms of the sequence:

1st term (a1) 18 2nd term (a2) 18(1/3) 6 3rd term (a3) 6(1/3) 2 4th term (a4) 2(1/3) 2/3 5th term (a5) 2/3(1/3) 2/9 6th term (a6) 2/9(1/3) 2/27

The sequence is 18, 6, 2, 2/3, 2/9, 2/27, and so on.

Conclusion

The common ratio of the geometric progression where the 6th term is 2/27 and the first term is 18 is 1/3. This is a fundamental concept in understanding geometric sequences, which is widely applicable in various fields from physics to economics.

By mastering the process of determining the common ratio, students and professionals can tackle more complex problems in mathematics and related disciplines. The method detailed in this article can be extended to find common ratios in other geometric progressions where the first term and a specific term are known.