(B) For the expansion of $(x + a)^n$,if $n$ is even,the middle term is the $(\frac{n}{2} + 1)$-th term.Here,$n = 10$,which is even.Therefore,the middl

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(B) For the expansion of $(x + a)^n$,if $n$ is even,the middle term is the $(\frac{n}{2} + 1)$-th term.Here,$n = 10$,which is even.Therefore,the middle term is the $(\frac{10}{2} + 1) = 6$-th term,denoted as $T_6$.The general term $T_{r+1}$ in the expansion of $(x + a)^n$ is given by $T_{r+1} = ^nC_r x^{n-r} a^r$.For $T_6$,we have $r = 5$.Substituting the values,$T_6 = ^{10}C_5 (x)^{10-5} (\frac{1}{x})^5$.$T_6 = ^{10}C_5 x^5 \cdot \frac{1}{x^5} = ^{10}C_5$.Thus,the middle term is $^{10}C_5$.

Class 11 Mathematics Binomial Theorem General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

Easy

The middle term in the expansion of ${\left( x + \frac{1}{x} \right)^{10}}$ is

A
$^{10}C_4 \frac{1}{x}$
B
$^{10}C_5$
C
$^{10}C_5 x$
D
$^{10}C_7 x^4$

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Binomial Theorem

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General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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DifficultJEE MAIN 2023

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The term independent of $x$ in the expansion of $\left( 9x - \frac{1}{3\sqrt{x}} \right)^{18}, x > 0$,is $\alpha$ times the corresponding binomial coefficient. Then $\alpha$ is:

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