The middle term in the expansion of x-1 - 18 is

X 1x18 Since 18 is even there is only one middle term i.e. 2n2+1thterm i.e. (n + 1)th term i.e. #160;Tn+1= #160;2nCn(x)2n n #160; 1xn= #160;2nCn #160; ...

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Standard XII

Mathematics

nth Term of A.P

The middle te...

Question

The middle term in the expansion of ${(x - \frac{1}{x})}^{18}$ is ___________.

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Solution

${(x - \frac{1}{x})}^{18}$
Since 18 is even
there is only one middle term i.e. ${(\frac{2 n}{2} + 1)}^{th}$ term
i.e. (n + 1)^th term
$\begin{array}{rcl} i . e . T_{n + 1} & = & ^{2 n} C_{n} {(x)}^{2 n - n} {(\frac{- 1}{x})}^{n} \\ = & ^{2 n} C_{n} {(- 1)}^{n} x^{2 n - n} x^{- n} \\ = & ^{2 n} C_{n} {(- 1)}^{n} x^{0} i . e .^{18} C_{9} {(- 1)}^{18} =^{18} C_{9} \end{array}$

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The middle term in the expansion of x-1x18 is ___________.

x-1x18 Since 18 is even there is only one middle term i.e. 2n2+1thterm i.e. (n + 1)th term i.e. Tn+1= 2nCn(x)2n-n -1xn= 2nCn (-1)n x2n-n x-n= 2nCn (-1)n x0 i.e. 18C9 (-1)18= 18C9

The middle term in the expansion of ${(x - \frac{1}{x})}^{18}$ is ___________.