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Question
(1+x)n−nx−1 is divisible by (where n ϵN)
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Solution
The correct option is B
(1+x)n=1+nx+n(n−1)2!x2+n(n−2)3!x3+⋯
⇒(1+x)n−nx−1
=x2[n(n−1)2!+n(n−1)(n−2)3!x+⋯]
From above it is clear that (1+x)n−nx−1 is divisible by x2
Trick : (1+x)n−nx−1. Put n=2 and x=3; Then 42−2.3−1=9 is not divisible by 6, 54 but divisible by 9.
Which is given by option (b) i.e., x2=9
(1+x)n=1+nx+n(n−1)2!x2+n(n−2)3!x3+⋯
⇒(1+x)n−nx−1
=x2[n(n−1)2!+n(n−1)(n−2)3!x+⋯]
From above it is clear that (1+x)n−nx−1 is divisible by x2
Trick : (1+x)n−nx−1. Put n=2 and x=3; Then 42−2.3−1=9 is not divisible by 6, 54 but divisible by 9.
Which is given by option (b) i.e., x2=9
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