3 real numbers that form a geometric progression have sum equal to 175 and product equal to 17576. What is the sum of the largest and smallest numbers?

Answer 1

a + ar + ar^2 = 175
a(1 + r + r^2) = 175

a(ar)(ar^2) = 17576
a^3 r^3 = 17576
(ar)^3 = 17576
take cube root
ar = 26
then
a(1+r+r^2)/ar = 175/26
(1 + r + r^2)/r = 175/26
26r^2 + 26r + 26 = 175r
26r^2 - 149r + 26 = 0
r = (149 ± √19497)/52 = appr 5.5506.. (I stored it)
then a = 26/5.5506 = appr 4.68417

the smallest number is 4.68417
and the largest is 144.3158

or

r = (149-√19497)/52 = appr .18016
a = 26/.18016 = appr 144.3158

the 3 numbers from largest to smallest are
144.3158 , 26 , and 4.684

notice the numbers are the same.

check:
144.3158 + 26 + 4.684 = 174.9998 , not bad
(144.3158)(26)(4.684) = 171757.355 , close enough