Real roots of equation 2x3-19x2+57x=k are first three terms of geometric progression then 1)value of k is a)216 b)108 c)-54 d)-108 2)if geometric progression is increasing sum of first n terms is(n in each option is raise to) a)(3/2)n-1 b)4((3/2)n-1) c)6((3/2)n-1) d)4(2n-1) 3if geometric progression is decreasing sum of infinite no. of terms is a)27/2 b)9 c)9/2 d)12

Hello student,
Please find the answer to your question below
Given equation is 2x³-19x²+57x-k=0
Also given they are in GP so let the terms be a/r,a,ar
as the product of roots is k so k=a³
By using the formula sum of roots=a/r+a+ar=19/2
Sum of product of roots=57/2
and product of roots=k/2=a³
by solving above equations we get a=-3,
So k=-54
Similarly by applying the formula we getif geometric progression is increasing sum of first n terms is a(rⁿ-1)/r-1=4((3/2)ⁿ-1)
Similarly by applying the formula we getif geometric progression is decreasing sum of infinite no. of terms is=a/1-r=27/2