Let the coefficients of the third,fourth,and fifth terms in the expansion of $(x + \frac{a}{x^2})^n, x \neq 0,$ be in the ratio $12 : 8 : 3$. Then the term independent of $x$ in the expansion is equal to ...... .

If the coefficients of $x^3$ and $x^4$ in the expansion of $(1 + ax + bx^2)(1 - 2x)^{18}$ in powers of $x$ are both zero,then $(a, b)$ is equal to

If the coefficient of $x^{15}$ in the expansion of $(ax^3 + \frac{1}{bx^{1/3}})^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $(ax^{1/3} - \frac{1}{bx^3})^{15}$,where $a$ and $b$ are positive real numbers,then for each such ordered pair $(a, b):$

If the coefficients of $x^{7}$ and $x^{8}$ in the expansion of $(2+\frac{x}{3})^{n}$ are equal,then the value of $n$ is equal to $.....$

If the term independent of $x$ in the expansion of $\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{9}$ is $k,$ then $18 k$ is equal to

The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of $(x^2 + \frac{2}{x})^{15}$ is