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(1+x)^n-n x-1  is divisible by (where n \in N )

 

Option: 1

2 x


Option: 2

x^2


Option: 3

2 x^3


Option: 4

All of these


Answers (1)

(1+x)^n=1+n x+\frac{n(n-1)}{2 !} x^2+\frac{n(n-1)(n-2)}{3 !} x^3+\ldots . . \\

\Rightarrow(1+x)^n-n x-1 \\

=x^2\left[\frac{n(n-1)}{2 !}+\frac{n(n-1)(n-3)}{3 !} x+\ldots . .\right]

From above it is clear that  (1+x)^n-n x-1  is divisible by x^2.

Trick : (1+x)^n-n x-1. Put n=2 and x=3; Then 4^2-2.3-1=9  is not divisible by 6, 54 but divisible by 9 . Which is given by option (b) i.e., x^2=9

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Sumit Saini

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