Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of these three terms is a, and b = 10. If a > c, what is the product of the first 4 terms of this G.P.?

Suraj Pandey

Understand the problem
Given: a, b, c in G.P.
Median is a, and b = 10
|a + b + c| = 15
a > c

Express G.P. terms
Since b is the middle term: a/r, 10, ar
|(a/r) + 10 + ar| = 15

Solve the absolute value equation
Case 1: (a/r) + 10 + ar = 15
(a/r) + ar = 5
a(r + 1/r) = 5

Case 2: (a/r) + 10 + ar = -15
(a/r) + ar = -25
a(r + 1/r) = -25

Solve for r
Let's solve case 1 since |a + b + c| should be positive.
ar = k, where k is the common ratio.

a(1/r + r) = 5
10(1/r + r) = 5
1/r + r = 0.5
Let r^2 + 2 = k, then
2 + 1/r = 5

Determine the product of first 4 terms
First 4 terms: a/r, 10, ar, a(r^2)

Since a > c:
a > ar
ar = 10, hence, c = a

Product of first 4 terms: (a/r) * 10 * ar * a