Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x\,>\,0$
Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x\,>\,0$
We have ${T_{r + 1}} = {\,^{18}}{C_r}{(\sqrt[3]{x})^{18 - r}}{\left( {\frac{1}{{2\sqrt[3]{x}}}} \right)^r}$
$ = {\,^{18}}{C_r}{x^{\frac{{18 - r}}{3}}} \cdot \frac{1}{{{2^r} \cdot {x^{\frac{r}{3}}}}} = {\,^{18}}{C_r}\frac{1}{{{2^r}}} \cdot {x^{\frac{{18 - 2r}}{3}}}$
Since we have to find a term independent of $x$, i.e., term not having $x$, so take $\frac{18-2 r}{3}=0$
We get $r=9 .$ The required term is ${\,^{18}}{C_9}\frac{1}{{{2^9}}}$
Similar Questions
If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :
If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :
- [JEE MAIN 2024]
If $^n{C_{r - 2}} = 36$ , $^n{C_{r - 1}} = 84$ and $^n{C_r} = 126$ , then value of $^n{C_{2r}}$ is
If $^n{C_{r - 2}} = 36$ , $^n{C_{r - 1}} = 84$ and $^n{C_r} = 126$ , then value of $^n{C_{2r}}$ is
The number of integral terms in the expansion of ${({5^{1/2}} + {7^{1/6}})^{642}}$ is
The number of integral terms in the expansion of ${({5^{1/2}} + {7^{1/6}})^{642}}$ is
The coefficient of $x^9$ in the expansion of $(1+x)\left(1+x^2\right)\left(1+x^3\right) \ldots . .\left(1+x^{100}\right)$ is
The coefficient of $x^9$ in the expansion of $(1+x)\left(1+x^2\right)\left(1+x^3\right) \ldots . .\left(1+x^{100}\right)$ is
- [IIT 2015]
The coefficients of three successive terms in the expansion of ${(1 + x)^n}$ are $165, 330$ and $462$ respectively, then the value of n will be
The coefficients of three successive terms in the expansion of ${(1 + x)^n}$ are $165, 330$ and $462$ respectively, then the value of n will be