The middle term in the expression of ${\left( {x - \frac{1}{x}} \right)^{18}}$ is

${r^{th}}$ term in the expansion of ${(a + 2x)^n}$ is

Coefficient of $x$ in the expansion of ${\left( {{x^2} + \frac{a}{x}} \right)^5}$ is

The value of $x$ in the expression ${[x + {x^{{{\log }_{10}}}}^{(x)}]^5}$, if the third term in the expansion is $10,00,000$

Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x\,>\,0$

In the expansion of $\left(\frac{\mathrm{x}}{\cos \theta}+\frac{1}{\mathrm{x} \sin \theta}\right)^{16},$ if $\ell_{1}$ is the least value of the term independent of $x$ when $\frac{\pi}{8} \leq \theta \leq \frac{\pi}{4}$ and $\ell_{2}$ is the least value of the term independent of $x$ when $\frac{\pi}{16} \leq \theta \leq \frac{\pi}{8},$ then the ratio $\ell_{2}: \ell_{1}$ is equal to