1+(1/3) x+(1.4/3.6) x2+(1.4 .7/3.6 .9) x3+⋅s is equal to

Question

1+(1/3) x+(1.4/3.6) x2+(1.4 .7/3.6 .9) x3+⋅s is equal to (A) x (B) (1+x)1 / 3 (C) (1-x)1 / 3 (D) (1-x)-1 / 3

Tardigrade Admin · Accepted Answer

Let (1+y)n=1+(1/3) x+(1 ⋅ 4/3 ⋅ 6) x2+(1 ⋅ 4 ⋅ 7/3 ⋅ 6 ⋅ 9) x3+⋅s =1+n y+(n(n-1)/2 !) y2+ ldots Comparing the terms, we get n y=(1/3) x, (n(n-1)/2 !) y2=(1 ⋅ 4/3 ⋅ 6) x2 Solving, n=-(1/3), y=-x Hence, given series =(1-x)-1 / 3.

Q. $1 + \frac{1}{3} x + \frac{1.4}{3.6} x^{2} + \frac{1.4.7}{3.6.9} x^{3} + \dots$ is equal to

$x$

$(1 + x)^{1/3}$

$(1 - x)^{1/3}$

$(1 - x)^{- 1/3}$

Q. 1+31​x+3.61.4​x2+3.6.91.4.7​x3+⋯ is equal to

x

(1+x)1/3

(1−x)1/3

(1−x)−1/3

Let (1+y)n=1+31​x+3⋅61⋅4​x2+3⋅6⋅91⋅4⋅7​x3+⋯ =1+ny+2!n(n−1)​y2+… Comparing the terms, we get ny=31​x,2!n(n−1)​ y2=3⋅61⋅4​x2 Solving, n=−31​,y=−x Hence, given series =(1−x)−1/3.

Q. $1 + \frac{1}{3} x + \frac{1.4}{3.6} x^{2} + \frac{1.4.7}{3.6.9} x^{3} + \dots$ is equal to

$x$

$(1 + x)^{1/3}$

$(1 - x)^{1/3}$

$(1 - x)^{- 1/3}$