Geometric progression (GP) – find 4th term: In a GP, the 5th term is 81 and the first term is 16. Determine the 4th term.

Introduction / Context:
Given two terms of a geometric progression (GP), we can find the common ratio and then any required term using the standard formula a_n = a * r^(n−1).

Given Data / Assumptions:

• a (first term) = 16
• a_5 = 81
• GP term formula: a_n = a * r^(n−1)

Concept / Approach:
From a_5 = a * r^4, compute r^4 = 81/16. Recognize 81 = 3^4 and 16 = 2^4 ⇒ r = 3/2 (taking the positive ratio for a standard increasing GP). Then a_4 = a * r^3.

Step-by-Step Solution:
r^4 = 81/16 ⇒ r = 3/2.a_4 = 16 * (3/2)^3 = 16 * 27/8 = 2 * 27 = 54.

Verification / Alternative check:
a_5 = 16 * (3/2)^4 = 16 * 81/16 = 81, consistent with the given data.

Why Other Options Are Wrong:
36, 24, 18, 48 arise from incorrect powers or mistaken ratio extraction.

Common Pitfalls:
Using r^5 instead of r^4; forgetting that the 4th term uses r^3, not r^4.

Final Answer:
54