Question

$the middle term of (x + \frac{1}{x} )^{2n}$
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Answer 1

Topic :-

Binomial Theorem

Given :-

$\mathtt{\left ( x+\dfrac{1}{x}\right )^{2n}}$

To Find :-

Middle term of given expression.

Concept Used :-

The middle term in the expansion of (x + y)ⁿ, when 'n' is even is given by :-

$\mathtt{T_{(n+2)/2}=\:^nC_{n/2}.x^{n/2}.y^{n/2}}$

Note : There is only one middle term.

The middle term in the expansion of (x + y)ⁿ, when 'n' is odd is given by :-

$\mathtt{T_{(n+1)/2}\:and\:T_{[(n+1)/2]+1}}$

Note : There are two middle terms.

Solution :-

2n is an even number. So, there will be only one middle term.

Here, we will have to replace :-

• 'x' with 'x'
• 'y' with '1/x' and
• 'n' with '2n' to apply the concept.

Applying the concept,

$\mathtt{T_{(2n+2)/2}=\:^{2n}C_{2n/2}.(x^{2n/2}).\left(\dfrac{1}{x}\right)^{2n/2}}$

$\mathtt{T_{(n+1)}=\:^{2n}C_{n}.(x^{n}).\left(\dfrac{1}{x}\right)^{n}}$

$\mathtt{T_{(n+1)}=\:^{2n}C_{n}.\left(\dfrac{x}{x}\right)^{n}}$

$\mathtt{T_{(n+1)}=\:^{2n}C_{n}.1^{n}}$

$\mathtt{T_{(n+1)}=\:^{2n}C_{n}}$

Answer :-

So, the middle term of given expression is $\mathtt{\bold{\:^{2n}C_{n}}}.$

Answer 2

Given (x - 1/x)2n Since the index is 2n , which is even. So , there is only one middle term, i.e., (2n/2 + 1)th term=(n+1)th term