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  • In  is an even positive integer, then the condition that the greatest term in the expansion of <img alt="(1+x)^n" sr

In n is an even positive integer, then the condition that the greatest term in the expansion of (1+x)^n many have the greatest coefficient also, is

Option: 1

\frac{n}{n+2}<x<\frac{n+2}{n}


Option: 2

\frac{n+1}{n}<x<\frac{n}{n+1}


Option: 3

\frac{n}{n+4}<x<\frac{n+4}{4}


Option: 4

None of these


Answers (1)

best_answer

If n is even then greatest coefficient  ={ }^n C_{n / 2}={ }^{2 m} C_m \quad (where n=2m)
$$ =(m+1) \text { th term }=T_{m+1}
Now, since T_{m+1} is greatest term
\begin{aligned} & \quad T_{m+1}>T_m ; T_{m+1}>T_{m+2} \\ & \Rightarrow \quad{ }^{2 m} C_m x^m>{ }^{2 m} C_{m-1} x^{m-1},{ }^{2 m} C_m x^m>{ }^{2 m} C_{m+1} x^{m+1} \\ & \text { Simplifying the inequalities, we get } x>\frac{m}{m+1}, x<\frac{m+1}{m} \quad \text { or } \quad x>\frac{n}{n+2} \text { and } x<\frac{n+2}{n} \end{aligned}

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