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

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 $\left( {\dfrac{{n + 1}}{2}} \right)$.