Question

Prove that there is no term involving x^5 int he expansion of (2x^2-3/x)^11

Answer 1

$\text{Consider,}$

$(2x^2-\frac{3}{x})^{11}$

$\textbf{Concept:}$

$\textbf{The general term in the expansion of (a+b)^n is}$

$\bf\;T_{r+1}=^nC_r\;a^{n-r}\;b^r$

$T_{r+1}=^{11}C_r\;(2x^2)^{11-r}\;(-\frac{3}{x})^r$

$T_{r+1}=^{11}C_r\;2^{11-r}x^{22-2r}\;(\frac{(-3)^r}{x^r})$

$T_{r+1}=^{11}C_r\;2^{11-r}x^{22-3r}\;(-3)^r$

$T_{r+1}=^{11}C_r\;2^{11-r}\;(-3)^r\,x^{22-3r}$

$\text{Let T_{r+1} be the term containing x^5 }$

$\implies\,22-3r=5$

$\implies\,3r=17$

$\implies\,r=\frac{17}{3}\text{ is not an integer}$

$\text{Hence, there is no term containing x^5 }$