SOLUTION: Find four numbers in GP whose sum is 85 and their product is 4096

Question 483278: Find four numbers in GP whose sum is 85 and their product is 4096
Answer by mathstutor494(120) (Show Source): You can put this solution on YOUR website!
Let the 4 numbers in GP be
a/r, a, ar, ar^2
Given that sum is 85 and their product is 4096.
Accordingly....
a/r+a+ar+ar^2=85 & a/r*a*ar*ar^2=4096
a/r(1+r+r^2+r^3)=85 & a^4r^2 = 4096
Since a^4r^2 = 4096 so a^4= 4096/r^2 = 8^4/r^2
So a = 8/sqrt(r)
Substituting value of a in sum equation leads to ...
(8/sqrt(r)r)(1+r+r^2+r^3)=85
(1+r+r^2+r^3)/(sqrt(r)r)= 85/8
Therefore
(1+r+r^2+r^3)= 85 and sqrt(r)r = 8
Solving 2nd eqn
r*r^(1/2)= 2^(3/2)
r^(3/2)= 4^(3/2)
so r = 4
Substituting value of r in below eqn and solving for a
a^4r^2 = 4096
(a^4)(4^2)=4096
(a^4)*16= 4096
(a^4)= 4096/16 = 256 = 4^4
So a = 4
Since four numbers in GP are...a/r, a, ar, ar^2
Therefore four numbers in GP are...
4/4, 4, 4*4, 4*4^2
1, 4, 16, 64

RELATED QUESTIONS
Find four numbers in G.P whose sum is 85 and product is... (answered by greenestamps)

find 4 nos in gp whose sum is 85 & product is... (answered by richwmiller,MathTherapy)

Find three numbers in a GP whose sum is 26 and and product is... (answered by edjones,richwmiller)

Find three numbers in GP such that their sum is 39 and their product is... (answered by Alan3354,greenestamps)

sum of infinite gp is 2 and sum of their cubes of the terms is 8/7 find... (answered by math_helper)

Find two positive real numbers whose sum is 70 and their product is... (answered by Alan3354)

find three numbers in arithmetic progression whose sum is 21 and whose product is... (answered by stanbon,RAY100)

find three numbers in arithmetic progression whose sum is 21 and whose product is... (answered by solver91311)

Find 3 numbers in the geometrical progression whose sum is 28 and whose product is... (answered by ikleyn)