Question

If the continued product of three numbers in G.P. is 216 and the sum of their products in pairs is 156, find the numbers.

Answer 1

Let the numbers are `a/r,a and ar.`
Then, `a/r*a*ar = 216`
`=>a^3 = 216 => a= 6`
Also,
`a/r(a)+a(ar)+a/r(ar) = 156`
`=> a^2(1/r+r+1) = 156`
`=> (1+r+r^2)/r = 156/a^2`
`=> (1+r+r^2)/r = 156/36`
`=> (1+r+r^2)/r = 13/3`
`=>3+3r+3r^2 -13r = 0`
`=>3r^2-10r+3 = 0`
`=>3r^2-9r-r+3 = 0`
`=>(3r-1)(r-3) =0`
`=> r = 1/3 or r = 3`
So, the three numbers are , `2,6 and 18.`