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(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

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

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(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 $T_{m+3}$.

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