Question

If sum of three numbers in G.P. is 38, and their product is 1728, find the numbers.

Answer 1

let the numbers are a/r, a and ar
product = 1728
⇒ a/r * a * ar = 1728
⇒ a³ = 1728
⇒ a = 12

12/r + 12 + 12r = 38
⇒ 12 + 12r + 12r² = 38r
⇒ 12r² - 26r + 12 = 0
⇒ 6r² - 13r + 6 = 0
r = 2/3 and 3/2

numbers are  8, 12 and 18

Answer 2

a + ar + ar² = 38

a(1+r+r²) = 38 ----(1)

(a)(ar)ar² = 1728

a³r³ = 1728

(ar)³ = 1728

ar = 12

➺ r = 12/a

putting this value in equation (1)

a( 1+12/a + 144/a²) = 38

➺ a² + 12a + 144 = 38a

➺ a²-26a + 144 = 0

➺ (a-18)(a-8) = 0


➺ a = 18 & a = 8

➺ r = 2/3 & r = 3/2

r = 12/a

putting this we get

➺ a(1+12/a + 144/a²) = 38

➺ a²-26a + 144 = 0

➺(a-18)(a-8) = 0

➺ a = 18 & a = 8

➺ r = 2/3 & r = 3/2

So the numbers are 18 , 12 & 8