Embibe Experts Solutions for Exercise 5: Assignment

Expand $(2+3x{)}^{-5}$ up to four terms in descending powers of $x$.

By using the binomial expansion, expand ${\left(1+x+x^2\right)}^3$

Find $6^{\mathrm{th} }$ term in the expansion of ${\left(\frac{4x}{5}-\frac{5}{2x}\right)}^9$

Find the middle terms in the expansion of ${\left(4x-\frac{x^3}{8}\right)}^7$.

Find the coefficient of $\frac{1}{y^2}$ in the expansion of ${\left(y+\frac{c^3}{y^2}\right)}^{10}$.

Find the value of $k$ so that the term independent of $x$ in the expansion of ${\left(\sqrt{x}+\frac{k}{x^2}\right)}^{10}$ is $405$ .

If the coefficient of $a^{r-1},a^r,a^{r+1}$ in the binomial expansion of $(1+a{)}^n$ are in A.P. then prove that $n^2-n(4r+1)+4r^2-2=0$

In the expansion of ${\left(1-x\right)}^{2n-1}, a_r$ is denoted as the coefficient of $x^r$, then prove that $a_{r-1}+a_{2n-r}=0$.