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  • If n is even positive integer, then the condition that the greatest term in the expansion of &

If n is even positive integer, then the condition that the greatest term in the expansion of (1+x)^n may 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, the greatest coefficient is { }^n C_{n / 2}

Therefore the greatest term ={ }^n C_{n / 2} x^{n / 2}

\begin{aligned} & \therefore{ }^n C_{n / 2} x^{n / 2}>{ }^n C_{(n / 2)-1} x^{(n-2) / 2} \\ & \text { and }{ }^n C_{n / 2} x^{n / 2}>{ }^n C_{(n / 2)+1} x^{(n / 2)+1} \end{aligned}

\begin{aligned} & \frac{n-\frac{n}{2}+1}{\frac{n}{2} x>1} \text { and } \frac{\frac{n}{2}}{\frac{n}{2}+1} x<1 \\ & x>\frac{\frac{n}{2}}{\frac{n}{2}+1} \text { and } \quad x<\frac{\frac{\frac{n}{2}+1}{\frac{n}{2}}}{n} \\ & \Rightarrow \quad x>\frac{n}{n+2} \text { and } \quad x<\frac{n+2}{n} \\ & \end{aligned}

Posted by

Gautam harsolia

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