if the sum of odd terms and the sum of even terms in the expansion of ${(x+a)}^n$ are $p\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}q$ respectively then $p^2+q^2=$

$p=$sum of odd terms $={}^n{\text{C}}_0{x}^n+{}^n{\text{C}}_2{x}^{n-2}{a}^2+\cdots$

$q=$ sum of even terms $={}^n{\text{C}}_1{x}^{n-1}a+{}^n{\text{C}}_3{x}^{n-3}{a}^3+\cdots$

$\begin{array}{l}{p}^2+q^2=\frac{1}{2}[{(p+q)}^2+{(p-q)}^2]\\ =\frac{1}{2}[{\left({(x+a)}^n\right)}^2+{\left({(x-a)}^n\right)}^2]\\ =\frac{1}{2}[{(x+a)}^{2n}+{(x-a)}^{2n}]\end{array}$