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.?

Question

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 · Accepted Answer

Understand the problemGiven: a, b, c in G.P.Median is a, and b = 10|a + b + c| = 15a > c
Express G.P. termsSince b is the middle term: a/r, 10, ar|(a/r) + 10 + ar| = 15
Solve the absolute value equationCase 1: (a/r) + 10 + ar = 15(a/r) + ar = 5a(r + 1/r) = 5
Case 2: (a/r) + 10 + ar = -15(a/r) + ar = -25a(r + 1/r) = -25
Solve for rLet's solve case 1 since |a + b + c| should be positive.ar = k, where k is the common ratio.
a(1/r + r) = 510(1/r + r) = 51/r + r = 0.5Let r^2 + 2 = k, then2 + 1/r = 5
Determine the product of first 4 termsFirst 4 terms: a/r, 10, ar, a(r^2)
Since a > c:a > arar = 10, hence, c = a
Product of first 4 terms: (a/r) * 10 * ar * a