Find the three numbers in geometric progression(G.P.) whose sum is 26 and product is 216.

Find the three numbers in geometric progression(G.P.) whose sum is 26 and product is 216.

Solution:

Let us assume that the three numbers in geometric progression are a/r,a and ar.
Given,product is 216.
So, (a/r)*a*ar=216
⇒a³ =216
⇒a³=6³
⇒a=6
Also given that
(a/r)+a+ ar=26
⇒(6/r)+6+6r=26
⇒(6/r)+6r =26-6
⇒(6/r)+6r=20
⇒(6+6r²)/r=20  (by taking the LCM)
⇒6+6r²=20r
⇒3+3r²=10r  (dividing the equation by 2)
⇒3r²-10r+3=0
⇒3r²-(9+1)r+3=0
⇒3r²-9r-r+3=0
⇒3r(r-3)-(r-3)=0
⇒(r-3)(3r-1)=0
either r-3=0⇒r=3
or, 3r-1=0⇒3r=1⇒r=1/3
if r=3 then the numbers are: 6/3,6, 6*3 ie 2,6,18.
If r=1/3 then the numbers are: 6/(1/3),6,6*(1/3) ie 6*3,6,6/3 ie 18,6,2.