General Term and Middle Term

Find the term independent of $x$ in the expansion of ${\left[\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{2x^2}\right]}^{10}$

Find the coefficient of $x^{-5}$ in ${}^{n}C_0-{}^{n}C_1\left(\frac{2x-1}{x}\right)+{}^{n}C_2{\left(\frac{2x-1}{x}\right)}^2+....+{\left(-1\right)}^n{\left(\frac{2x-1}{x}\right)}^n$

If the $2$nd, $3$rd and $4$th terms in the expansion of ${\left(a+x\right)}^n$ are $240, 720, 1080$ respectively, find the values of $a,x$ and $n$.

If the coefficients of the $p$th, $\left(p+1\right)$th and $\left(p+2\right)$th terms in the expansion of ${\left(1+x\right)}^n$ are in A.P. prove that $n^2-\left(4p+1\right)n+4p^2-2=0$.

If the ratio of the fifth terms from the beginning and from the end in the expansion of ${\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)}^n$ is $\sqrt{6}:1$, then find the value of $n$.

If the ratio of the seventh term from beginning and the seventh term from the end in the expansion of ${\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)}^n$ is $1:6$, then find the value of $n$.

If the ratio of the coefficients of $\left(r-1\right)$th, $r$-th and $\left(r+1\right)$th terms in the expansion of ${\left(1+x\right)}^n$ is $1:3:5$, then find the values of $n$and $r$.

If the coefficients of three consecutive terms in the expansion of ${\left(1+x\right)}^n$ are $76, 95$ and $76$ respectively, then find the value of $n$.

If the sum of the coefficients of first three terms in the expansion of ${\left(x-\frac{3}{x^2}\right)}^n$ is $559$, then find the coefficient of $x^3$.

If the coefficients of the $p$-th and $q$-th terms $(p\ne q)$ in the expansion of ${(1+x)}^n$ are equal, then show that $p+q=n+2$

Prove that the middle term of ${\left(1+x\right)}^{2n}$ is $\frac{1·3·5 .....\left(2n-1\right)2^n{x}^n}{n!}$.

Prove that the middle term of ${\left(x+\frac{1}{2x}\right)}^{2n}$ is $\frac{1·3·5 .....\left(2n-1\right)}{n!}$

Find the middle term/terms in the expansion of  ${\left(x^{\frac{1}{3}}-\frac{1}{2}{x}^{-\frac{1}{2}}\right)}^6$

Find the middle term/terms in the expansion of  ${\left(\frac{\mathrm{x}}{\mathrm{y}}-\frac{\mathrm{y}}{\mathrm{x}}\right)}^9$

Find the middle term/terms in the expansion of  ${\left(\frac{x}{a}+\frac{b}{x}\right)}^6$

Find the middle term/terms in the expansion of  ${\left(a+2b\right)}^4$

Find the middle term/terms in the expansion of  ${\left(2x+3\right)}^7$

Expand upto the term independent of $x$:

${\left(3x^2-\frac{1}{2x^3}\right)}^5$

Expand upto the term independent of $x$:

${\left(2x-\frac{1}{x^2}\right)}^6$

Expand upto the term independent of $x$:

${\left(x+\frac{1}{x}\right)}^6$