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Question: If the middle terms in the expansion of \[{({x^2} + \dfrac{1}{x})^{2n}}\] is \[184756{x^{10}}\], the...

If the middle terms in the expansion of (x2+1x)2n{({x^2} + \dfrac{1}{x})^{2n}} is 184756x10184756{x^{10}}, then what is the value of nn?
A. 1010
B. 88
C. 55
D. 66

Explanation

Solution

To solve this question expand the given expression. And then find the middle term of the given expression and then equate the power with power and the coefficient to the coefficient of the given value in the question. After solving further, find the value of nn. The value expression that is to expand is (x2+1x)2n{({x^2} + \dfrac{1}{x})^{2n}} this is in the form of (a+b)n{(a + b)^n}.

Complete step-by-step solution:
Given,
An expression to expand (x2+1x)2n{({x^2} + \dfrac{1}{x})^{2n}}
And the middle term of the expression is 184756x10184756{x^{10}}.
To find,
The value of nn
Formula used:
Expansion of (a+b)n{(a + b)^n}
(a+b)n=nC0anb0+nC1an−1b1+nC2an−2b2+...................+nCn−1a1bn−1+nCna0bn{(a + b)^n} = {n_{{C_0}}}{a^n}{b^0} + {n_{{C_1}}}{a^{n - 1}}{b^1} + {n_{{C_2}}}{a^{n - 2}}{b^2} + ................... + {n_{{C_{n - 1}}}}{a^1}{b^{n - 1}} + {n_{{C_n}}}{a^0}{b^n}
Here,
In the given question, aa is x2{x^2} and bb is 1x\dfrac{1}{x}
We have to find the middle term in the expansion.
Total power to the expression is 2n2n.
So, the total number of terms is one greater than power. That is 2n+12n + 1.
Middle term of the expression in the nth{n^{th}} term.
nth{n^{th}} term in the expression is
nth term=2nCnan+1bn−1{n^{th}}\,term = 2{n_{{C_n}}}{a^{n + 1}}{b^{n - 1}}
On putting the value of aa and bb the expression look like
nth term=2nCn(x2)n(1x)n{n^{th}}\,term = 2{n_{{C_n}}}{({x^2})^n}(\dfrac{1}{x})n
Onn further solving
nth term=2nCn(x2n−n){n^{th}}\,term = 2{n_{{C_n}}}({x^{2n - n}})
Now solving the power
nth term=2nCn(xn){n^{th}}\,term = 2{n_{{C_n}}}({x^n}) …………………………………(i)
This is also middle term
Now from the question middle term is
middle term=184756x10middle\,term = 184756{x^{10}} …………………………..……(ii)
On equating the values of middle term from equation 1 and 2
2nCn(xn)=184756x102{n_{{C_n}}}({x^n}) = 184756{x^{10}}
On comparing the power of x both side we get
xn=x10{x^n} = {x^{10}}
From here,
n=10n = 10
Final answer:
From her value of n satisfying the condition of middle term 184756x10184756{x^{10}} is
⇒n=10\Rightarrow n = 10

Note: To solve these types of questions you must know the expansion of different terms and after expanding the expression compare the given term of the question with the respective term of the expression. And compare both the terms to get the value of any variable.