Prove that the coefficient of $x^n$ in the expansion of $\frac{1}{1+x+x^2}$ is $1, 0, -1$ according as $n$ is of the form $3m, 3m-1$ or $3m+1$

In the expansion of $(a+b+c{)}^8$ find $\left(1\right)$ the number of terms, $\left(2\right)$ the sum of the coefficients of the terms

Prove that if $n$ be an even integer then, $\frac{1}{1!(n-1)!}+\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}+\dots +\frac{1}{(n-1)!1!}=\frac{2^{n-1}}{n!}$

If $C_0, C_2, C_2,\dots , C_n$ are the coefficient in the expansion of $(1+x{)}^n,$ when $n$ is a positive integer. Prove that,

$\left(1\right) C_0-C_1+C_2-C_3+\dots \dots +{\left(-1\right)}^r{C}_r={\left(-1\right)}^r\frac{\left(n-1\right)!}{r !·\left(n-r-1\right)!}$

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

$\left(3\right) C_0^2-C_1^2+C_2^2-C_3^2+\dots +{\left(-1\right)}^n{C}_n^2=0\text{ or }{\left(-1\right)}^{\frac{n}{2}}\text{·}{C}_{\frac{n}{2}.}$

accordingly as $n$ is even or odd

If $S_n$ denote the sum of first $n$ natural numbers. Prove that

$\left(1\right) (1-x{)}^{-3}=S_1+S_{2}x+S_3{x}^2+\dots \dots +S_n{x}^{n-1}+\dots$

$\text{(2) }2\left(S_1{S}_{2n}+S_2{S}_{2n-1}+\dots .+S_n{S}_{n+1}\right)=\frac{(2n+4)!}{5!\left(\right(2n-1)!}$

If $q_n=\frac{1·3·5·7\dots \left(2n-1\right)}{2·4·6·8\dots ·2n}$, prove that, ${\mathit{\text{q}}}_{2n+1}+q_1{q}_{2n}+q_2{q}_{2n-1}+\dots +q_{n-1}{q}_{n+2}+q_n{q}_{n+1}=\frac{1}{2}$

If $q_n=\frac{1·3·5·7\dots ·\left(2n-1\right)}{2·4·6·8\dots ·2n},$ prove that

$2\left\{q_{2n}-q_1{q}_{2n-1}+q_2{q}_{2n-2}+\dots +{\left(-1\right)}^{n-1}{q}_{n-1}{q}_{n+1}\right\}=q_n+{\left(-1\right)}^{n-1}{q}_n^2$​​​​​​​.

Find the sum of the products, two at a time of the coefficients in the expansion of ${\left(1+x\right)}^n$ when $n$ is a positive integer.