Question

If the \[rth\]term is the middle term in the expansion of \[{\left( {{x^2} - \dfrac{1}{{2x}}} \right)^{20}}\]then the\[\left( {r + 3} \right)th\]term is
A.\[{}^{20}{C_{_{14}}} - \dfrac{1}{{{2^{14}}}}.x\]
B.\[{}^{20}{C_{12}} - \dfrac{1}{{{2^{12}}}}.{x^2}\]
C.\[ - \dfrac{1}{{{2^{13}}}}.{}^{20}{C_7}x\]
D.None of these

Answer 1

Hint: Binomial expansion theorem is a theorem which specifies the expansion of any power \[{\left( {a + b} \right)^n}\]of a binomial \[\left( {a + b} \right)\]as a sum of products e.g. \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]. The number of in the expansion depends upon the raised positive integral power. If the raised power of expansion is\[n\]then the number of terms of the expansion will be \[n + 1\].


Complete step by step solution:
Given \[rth\] term is the middle term
In the given function \[{\left( {{x^2} - \dfrac{1}{{2x}}} \right)^{20}}\], the raised power is 20 which is an even number hence the expansion will have \[n + 1 = 20 + 1 = 21\]numbers of terms and one middle term
The middle term for \[n + 1 = 21\]numbers of terms will be \[\dfrac{{21 + 1}}{2} = 11\]and as given in the question \[rth\]term is the middle term, hence \[rth\] is the 11th term of expansion
\[{\left( {{x^2} - \dfrac{1}{{2x}}} \right)^{20}} = {T_1} + {T_2} + {T_3} + ........ + {T_{20}} + {T_{21}}\]
\[r = 11\]
Hence the \[\left( {r + 3} \right)th\] term will be \[ = 11 + 3 = 14th\] term in the expansion
Now expand the function for 14th term we get,
\[
  {\left( {{x^2} - \dfrac{1}{{2x}}} \right)^{20}} = {}^{20}{C_{13}}{\left( {{x^2}} \right)^{\left( {20 - 13} \right)}}{\left( { - \dfrac{1}{{2x}}} \right)^{13}} \\
   = {}^{20}{C_{13}}{\left( {{x^2}} \right)^7}{\left( { - \dfrac{1}{{2x}}} \right)^{13}} \\
   = - {}^{20}{C_{13}}{x^{14}}\left( {{2^{ - 13}} \times {x^{ - 13}}} \right) \\
   = - {}^{20}{C_{13}}{x^{14 - 13}}\left( {{2^{ - 13}}} \right) \\
   = - {}^{20}{C_{13}}x\left( {{2^{ - 13}}} \right) \\
 \]
Hence, the \[\left( {r + 3} \right)th\]term of the expansion is\[ - \dfrac{{{}^{20}{C_{13}}x}}{{\left( {{2^{13}}} \right)}}\]
So, option D is the right answer


Note: The total number of terms in the expansion of ${\left( {x + y} \right)^n}$ is $\left( {n + 1} \right)$.
If n is even, then the middle term is \[\left( {\dfrac{n}{2}} \right){\text{ and }}\left( {\dfrac{n}{2} + 1} \right)\] and if n is odd, then the middle term is \[\left( {\dfrac{{n + 1}}{2}} \right)\].