Question

The value of the first term of a G.P if its sum of three numbers is 7 and their

product is 8.* find the gp.

Answer 1

Explanation:

Assume that the 4 terms are a/r, a , ar, ar2.

Product of first 3 terms = a3 =8

=> a=2

Sum of last 3 terms :

a(1 + r + r2) = 14

=> r2 + r - 6 = 0

Solving the above equation r= -3 or 2

So the terms are

-2/3, 2, -6, 18

Or

1, 2, 4, 8

Answer 2

Given:

Sum of three terms in a GP = 7

Product of the numbers = 8

Find:

G.P

Solution:

Let the three numbers be a/r , a , ar where a is the first term and r is the common ratio.

So,

⟹ a/r + a + ar = 7 -- equation (1)

And,

⟹ (a/r) * a * ar = 8

⟹ a³ = 8

⟹ a = ³√8

⟹ a = 2

Substitute the value a in equation (1).

⟹ 2/r + 2 + 2r = 7

⟹ (2 + 2r + 2r²)/r = 7

⟹ 2 + 2r + 2r² = 7r

⟹ 2r² + 2r - 7r + 2 = 0

⟹ 2r² - 5r + 2 = 0

⟹ 2r² - 4r - r + 2 = 0

⟹ 2r(r - 2) - 1 (r - 2) = 0

⟹ (2r - 1)(r - 2) = 0

★ 2r - 1 = 0

⟹ 2r = 1

⟹ r = 1/2

★ r - 2 = 0

⟹ r = 2

If r = 1/2 ;

a/r = 2/(1/2) = 2 * 2 = 4

a = 2

ar = 2 * 1/2 = 1

If r = 2 ;

a/r = 2/2 = 1

a = 2

ar = 2 * 2 = 4

∴ Required GP is 4 , 2 , 1 or 1 , 2 , 4.

I hope it will help you.

Regards.