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