The product of three numbers in a G.P. is 216 and the sum of their products in pairs is 156. Find the numbers.
Answer: A
Let the numbers be \(a/r, a, ar\). Product is \((a/r)(a)(ar) = a^3 = 216\), so \(a=6\). Sum of products in pairs: \((a/r)(a) + a(ar) + (a/r)(ar) = a^2/r + a^2r + a^2 = 156\). Substitute a=6: \(36/r + 36r + 36 = 156 \Rightarrow 36/r+36r=120\). Divide by 12: \(3/r+3r=10 \Rightarrow 3+3r^2=10r \Rightarrow 3r^2-10r+3=0 \Rightarrow (3r-1)(r-3)=0\). So \(r=3\) or \(r=1/3\). If r=3, numbers are 2, 6, 18. If r=1/3, numbers are 18, 6, 2.