Question

Find three terms of G.P,whose sum is 14 and product is 64​

Answer 1

Let the three terms of a GP be ----- a/r, a, ar

a ----- first term

r ------ common ratio

given,

sum of three terms = 14

a/r + a + ar = 14 -------------------- equation 1

product of three terms = 64

a/r × a × ar = 64

a^3 = 64

a^3 = 4^3 (which is 4×4×4)

by comparing, we get

a = 4

put this value on equation 1

a/r + a + ar = 14

4/r + 4 + 4r = 14

4/r + 4r = 14 - 4

(4 + 4r^2) / r = 10

cross multiplication

4r^2 + 4 = 10r

4r^2 - 10r + 4 = 0

divide this quadratic equation by 2

then we get

2r^2 - 5r + 2 = 0

now split the middle term

2r^2 - 4r - r + 2 = 0

2r(r-2) - 1(r-2) = 0

(r-2)(2r-1) = 0

hence, r = 1/2 or 2

Since, we got two values for r, therefore we also get two GPs

three terms are.

for r = 2

a/r = 4/2 = 2

a = 4

ar = 4×2 = 8

so,

2,4,8 are three terms of GP

OR

for r = 1/2

a/r = 4/(1/2) = 4×2 = 8

a = 4

ar = 4×1/2 = 2

so,

8, 4, 2 are three terms