If the m-th term is the middle term in the ex | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The expansion of $\left(x^2-\frac{1}{2x}\right)^{20}$ contains $20+1 = 21$ terms.Since the number of terms is odd,the middle term is the $\left(\f

Vedclass · Accepted Answer

-${}^{20}C_{13} 2^{-13}$

Vedclass · Answer

(C) The expansion of $\left(x^2-\frac{1}{2x}\right)^{20}$ contains $20+1 = 21$ terms.Since the number of terms is odd,the middle term is the $\left(\frac{20}{2}+1\right)$-th term,which is the $11$-th term.Thus,$m = 11$.We need to find the coefficient of $T_{m+3} = T_{11+3} = T_{14} = T_{13+1}$.The general term $T_{r+1}$ in the expansion of $(a+b)^n$ is given by ${}^nC_r a^{n-r} b^r$.For $T_{13+1}$,we have $n=20$,$r=13$,$a=x^2$,and $b=-\frac{1}{2x}$.$T_{14} = {}^{20}C_{13} (x^2)^{20-13} \left(-\frac{1}{2x}\right)^{13}$$T_{14} = {}^{20}C_{13} (x^2)^7 \left(-\frac{1}{2^{13} x^{13}}\right)$$T_{14} = -{}^{20}C_{13} \cdot \frac{x^{14}}{2^{13} x^{13}} = -{}^{20}C_{13} \cdot 2^{-13} x$.Therefore,the coefficient of $T_{m+3}$ is $-{}^{20}C_{13} 2^{-13}$.

If the $m$-th term is the middle term in the expansion of $\left(x^2-\frac{1}{2x}\right)^{20}$,find the coefficient of $T_{m+3}$.

Similar Questions

Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2+(1+x)^3+\cdots+(1+x)^{49}+(1+mx)^{50}$ is $(3n+1)^{51}C_3$ for some positive integer $n$. Then the value of $n$ is

The number of integral terms in the expansion of $\{7^{(1/2)} + 11^{(1/6)}\}^{824}$ is equal to ...................

The terms containing $x^r y^s$ (for certain $r$ and $s$) are present in both the expansions of $(x+y^2)^{13}$ and $(x^2+y)^{14}$. If $\alpha$ is the number of such terms,then the sum $\alpha \sum_{r, s}(r+s) =$

The coefficient of the term independent of $x$ in the expansion of ${\left( {\sqrt {\frac{x}{3}} + \frac{3}{{2{x^2}}}} \right)^{10}}$ is

If the fourth term in the expansion of $(x+x^{\log _{2} x})^{7}$ is $4480$,then the value of $x$ where $x \in N$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module