Q. If P be the sum of odd terms and Q that of even terms in the expansion of${\left(x+a\right)}^n$, prove  that 
 $$\begin{gathered} \left(i\right) \left(P^2-Q^2\right)={\left(x^2-a^2\right)}^n \\ \left(ii\right) 4 PQ = \left[{\left(x+a\right)}^{2n}-{\left(x-a\right)}^{2n}\right] \\ \left(iii\right) 2\left(P^2+Q^2\right)=\left[{\left(x+a\right)}^{2n}+{\left(x-a\right)}^{2n}\right] \end{gathered}$$