Introduction

Understanding the properties of geometric progressions (GP) is crucial in many mathematical applications. One such problem involves finding the 8th term of a GP, given the first term and the common ratio. This article will guide you through this process and explore related concepts of both geometric and arithmetic progressions.

Geometric Progression Basics

A geometric progression (GP) 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. The formula for the (n)th term of a GP is:

{a}_{n} {a}_{1} {r}^{n - 1}

where ({a}_{1}) is the first term and (r) is the common ratio.

Problem: Finding the 8th Term of a GP

The first term of a GP is 4, and its common ratio is 2. We need to find the 8th term of this sequence.

Step-by-Step Solution

a_1  4r  2a_8  a_1 * r^{8-1}a_8  4 * 2^7a_8  4 * 128a_8  512

The 8th term of the GP is therefore 512.

Additional Practice and Understanding

Let's consider a few additional problems and theorems related to geometric and arithmetic progressions to deepen your understanding.

Geometric Progression with a Common Difference

Suppose we have a geometric progression where the first term and a common difference (d) are used as follows:

2, 2d, 22d, 23d, 24d, ...

The second term is (2d), the fourth term is (2^3d 8d), and the eighth term is (2^7d 128d).

The ratio of the fourth term to the second term is:

(frac{8d}{2d} 4)

Similarly, the ratio of the eighth term to the fourth term is:

(frac{128d}{8d} 16)

The common ratio (r) can be calculated as:

(r  frac{128d}{256d}  2)

The common ratio is (2), and this implies that the sequence of terms follows a geometric progression with a common ratio of (2).

First Term and Common Ratio in Relation to Each Other

Suppose the second, fourth, and eighth terms of an arithmetic progression (AP), which are also the first three terms of a geometric progression (GP), are given by:

2, 2d, 23d, 27d

The relationship between these terms is:

(frac{23d}{2d}  2)(frac{27d}{23d}  4)

The common ratio (r) can be found as follows:

(r  frac{27d}{23d}  frac{23d}{2d})

By simplifying, we get:

(r frac{27d - 23d}{23d - 2d} frac{4d}{2d} 2)

The common ratio of the geometric progression is (2), indicating a consistent multiplication by (2) to move from one term to the next.

Conclusion

In this article, we explored the process of finding the 8th term of a geometric progression given the first term and the common ratio. We also examined additional properties and relationships within both geometric and arithmetic progressions. Understanding these concepts is essential for solving more complex problems in mathematics.