Embibe Experts Solutions for Chapter: Binomial Theorem, Exercise 1: Exercise-1

If $C_0,C_1,C_2,\dots C_n$ are the binomial coefficients in the expansion of ${\left(1+x\right)}^n$ then prove that:

$\left(C_0+C_1\right)\left(C_1+C_2\right)\left(C_2+C_3\right)\left(C_3+C_4\right)\dots \dots .\left(C_{n-1}+C_n\right)=\frac{C_0{C}_1{C}_2\dots \dots \dots C_{n-1}{\left(n+1\right)}^n}{n!}$

If $C_0,C_1,C_2,\dots C_n$ are the binomial coefficients in the expansion of ${\left(1+x\right)}^n$ then prove that

$C_0-2C_1+3C_2-4C_3+\dots .+{\left(-1\right)}^n\left(n+1\right)C_n=0$

If $C_0,C_1,C_2,\dots C_n$ are the binomial coefficients in the expansion of ${\left(1+x\right)}^n$ then prove that

$4C_0+\frac{4^2}{2}·C_1+\frac{4^3}{3}{C}_2+\cdots \cdots +\frac{4^{n+1}}{n+1}{C}_n=\frac{5^{n+1}-1}{n+1}$

If $C_0,C_1,C_2,\dots C_n$ are the binomial coefficients in the expansion of ${\left(1+x\right)}^n$ then prove that

$\frac{2^2·C_0}{1.2}+\frac{2^3·C_1}{2·3}+\frac{2^4·C_2}{3·4}+\dots \dots +\frac{2^{n+2}·C_n}{\left(n+1\right)\left(n+2\right)}=\frac{3^{n+2}-2n-5}{\left(n+1\right)\left(n+2\right)}$

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+\dots \dots +C_n{x}^n$, prove that

$C_0{C}_r+C_1{C}_{r+1}+\dots \dots +C_{n-r}{C}_n=\frac{\left(2n\right)!}{\left(n+r\right)!\left(n-r\right)!}$

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+\dots \dots +C_n{x}^n$, prove that

$C_0^2-C_1^2+C_2^2-C_3^2+\dots \dots +{\left(-1\right)}^n{C}_n^2=0 \text{or} {\left(-1\right)}^{\frac{n}{2}}{C}_{\frac{n}{2}}$

Find the coefficient of $x^{100}$ in $\frac{3-5x}{{\left(1-x\right)}^2}$.

Sum of coefficients of odd powers of $x$ in expansion of ${\left(9x^2+x-8\right)}^6$.