$(1-{\mathrm{i}}^2)(1-\frac{{\mathrm{i}}^2}{{(1+\mathrm{i})}^2})(1-\frac{{\mathrm{i}}^2}{{(1+2\mathrm{i})}^2}).............$

$\Rightarrow (1-\mathrm{i})\mathrm{·}\frac{(1+\mathrm{i})\left(1\right)}{1(1+\mathrm{i})}\frac{(1+2\mathrm{i})}{(1+\mathrm{i})}\mathrm{·}\frac{(1+\mathrm{i})(1+3\mathrm{i})}{(1+2\mathrm{i})(1+2\mathrm{i})}.....$

$\Rightarrow (1-\mathrm{i})\frac{(1+\mathrm{ni})}{\left(1+(\mathrm{n}-1)\mathrm{i}\right)}=\frac{\mathrm{p}+\mathrm{iq}}{1+\mathrm{iq}}$

$\Rightarrow \frac{1+\mathrm{n}+(\mathrm{n}-1)\mathrm{i}}{1+(\mathrm{n}-1)\mathrm{i}}=\frac{\mathrm{p}+\mathrm{iq}}{1+\mathrm{iq}}$

⇒         q = n – 1 ;  p = n + 1

if    n = 9, p = 10, q = 8  

⇒         p + q = 18

if    n = 11, p = 12, q = 10

⇒         p + q = 22