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

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2 x)^{ n }$ be in the ratio $2: 5: 8$. Then the coefficient of the term, which is in the middle of these three terms, is $...........$.

Find the middle terms in the expansions of $\left(3-\frac{x^{3}}{6}\right)^{7}$

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n$ is $\sqrt{6}: 1$, then the third term from the beginning is:

In the expansion of $(1+a)^{m+n},$ prove that coefficients of $a^{m}$ and $a^{n}$ are equal.

Find the $4^{\text {th }}$ term in the expansion of $(x-2 y)^{12}$