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.