The Third and Fifth Terms of a Geometric Progression (GP)

Given a geometric progression (GP) where the third term is 9/2 and the fifth term is 81/8, the objective is to find the first term and the common ratio. This article will guide you through the process using the GP formula:

Tn arn-1

where:

Tn is the n-th term a is the first term r is the common ratio n is the term number

Step-by-Step Solution

Assign Variables and Form Equations:

For the third term:

T3 ar2 9/2 (Equation 1)

For the fifth term:

T5 ar4 81/8 (Equation 2)

Divide Equation 2 by Equation 1 to Eliminate a:

Dividing Equation 2 by Equation 1, we get:

ar4/ar2 81/8 / 9/2

Simplifying the left side:

r2 (81/8) × (2/9)

Calculating the right side:

r2 (81 × 2) / (8 × 9) 9/4

Solve for r:

Taking the square root of both sides:

r ±(3/2)

Thus, the common ratio can be either 3/2 or -3/2.

Substitute r back into Equation 1 to find a:

For r 3/2:

9/2 a(3/2)2

a 9/2 ÷ (9/4) 2

Since the solution for r -3/2 will yield the same first term due to the positive nature of division in these contexts, we have:

a 2, r ±(3/2).

Summary of Results

Therefore, the first term a 2 and the common ratio r 3/2 or -3/2.

Understanding how to solve for the first term and common ratio in a geometric progression is crucial in various fields, including mathematics, physics, and engineering, where sequences and series are used to model and analyze phenomena.