R S Aggarwal Solutions for Exercise 1: EXERCISE 6A

Find the term independent of $x$ in the expansion of ${\left(\frac{3x^2}{2}-\frac{1}{3x}\right)}^9$.

The coefficients of three consecutive terms in the expansion of $(1+x{)}^n$ are in the ratio $1: 7: 42$. Find $n$.

In the binomial expansion of $(a+b{)}^n,$ the coefficients of the $4^{th}$ and ${13}^{th}$ terms are equal to each other. Find $n$.

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

If $a$ and $b$ are distinct integers, prove that $\left(a^n-b^n\right)$ is divisible by $(a-b)$, whenever $n$ is a positive integer.

Using binomial theorem, prove that $\left(3^{2n+2}-8n-9\right)$ is divisible by $64,$ where $n$ is a positive integer.

Using binomial theorem, prove that $6^n-5n,$ when divided by $25,$ always leaves a remainder $1$, where $n\in N$.

Find the coefficient of $x^4$ in the expansion of $(1+x{)}^n(1-x{)}^n$. Deduce that $C_2=C_0{C}_4-C_1{C}_3+C_2{C}_2-C_3{C}_1+C_4{C}_0$.