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

If 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 \text { (where } n=2 m \text { ) }

                                =(m+1) \text { th term }=T_{m+1}

Now, since T_{m+1} is greatest term

\begin{aligned} & 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} \end{aligned}

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}

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