Find the term in the expansion of ${\left(2x-5\right)}^6$ which have
(i) Greatest binomial coefficient
(ii) Greatest numerical coefficient
(iii) Algebraically greatest coefficient
(iv) Algebraically least coefficient

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:

$\frac{\left(3·2-1\right)}{2}{C}_1+\frac{3^2·2^2-1}{2^2}{C}_2+\frac{3^3·2^3-1}{2^3}{C}_3+\dots \dots .+\frac{3^n·2^n-1}{2^n}{C}_n=\frac{2^{3n}-3^n}{2^n}$

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

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

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

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