Question

Hey !✌️

The sum of three numbers in GP is 39/10 and their product is 1. Find the numbers.

Answer 1

let 3 no's be a, ar, ar.r

a + ar + ar.r = 39/10

a( 1 + r + r.r ) = 3.9

Also a.ar.ar.r = 1

a^3 r^3 = 1

ar = 1

a = 1/r

(1 + r +r.r)/r = 39/10

10 + 10 r + 10 r.r - 39 r= 0
10 r.r -29 r + 10 = 0
10r.r -25 r -4 r +10=0
5r ( 2 r -5) - 2( 2r -5) = 0

r = 2/5, 5/2

so a = 5/2,2/5

Answer 2

Answer:

5/2, 1, 2/5 (or) 2/5,1,5/2.

Step-by-step explanation:

Let the three numbers in GP be a, ar, ar²

(i)

Given that Sum of three numbers is (39/10).

⇒ a + ar + ar² = (39/10)  

(ii)

Given that product of three numbers is 1.

⇒ a * ar * ar² = 1

⇒ (ar)³ = 1

⇒ ar = 1

⇒ a = (1/r)

Substitute a = (1/r) in (i), we get

⇒ (1/r) + (1/r) * r + (1/r) * r² = 39/10

⇒ 1/r + 1 + r = 39/10

⇒ 1 + r + r² = 39r/10

⇒ 10(1 + r + r²) = 39r

⇒ 10 + 10r + 10r² = 39r

⇒ 10 + 10r + 10r² - 39r = 0

⇒ 10r² - 29r + 10 = 0

⇒ 10r² - 25r - 4r + 10 = 0

⇒ 5r(2r - 5) -2(2r - 5) = 0

⇒ (5r - 2)(2r - 5) = 0

⇒ r = (2/5) (or) 5/2.

(iii)

When r = 2/5:

⇒ a = 5/2

When r = 5/2:

⇒ 2/5.

(iv)

When a = 5/2 and r = 2/5:

⇒ a = 5/2

⇒ ar = (5/2) * (2/5) = 1

⇒ ar² = (5/2) * (2/5)² = 2/5

When a = 2/5 and r = 5/2:

⇒ a = 2/5

⇒ ar = (2/5) * (5/2) = 1

⇒ ar² = (2/5) * (5/2)² = 5/2

Therefore, the numbers are 5/2, 1, 2/5 (or) 2/5, 1, 5/2.

Hope it helps!