The product of five positive numbers of a geometric progression is 243. The ratio of second largest term to fourth largest term is 16:1. Find the largest term.

To solve this problem, we need to use the properties of a geometric progression (GP). A 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.

Step 1: Understand the problem The product of five terms of a GP is 243. This means that if the terms are a/ar, a, ar, ar^2, ar^3 (where a is the first term and r is the common ratio), then (a/ar) * a * ar * ar^2 * ar^3 = 243. Simplifying this gives a^5 = 243, so a = 3 (since 3^5 = 243).

Step 2: Find the common ratio The ratio of the second largest term to the fourth largest term is 16:1. In our sequence, the second largest term is ar^2 and the fourth largest term is a/ar. So, ar^2 / (a/ar) = 16/1. Simplifying this gives r^3 = 16, so r = 2.5 (since 2.5^3 = 15.625 which is approximately 16).

Step 3: Find the largest term The largest term in our sequence is ar^3 = 3 * (2.5)^3 = 3 * 15.625 = 46.875.

So, the largest term in the sequence is 46.875.