The period of  $\tan ky+\sin ky$, where $k=1+4+9+\mathrm{......20}$ terms, is

 $K=1^2+2^2+3^2+\cdot \cdot \cdot \cdot \cdot 20termi$
 $K=\frac{n\left(n+1\right)\left(2n+1\right)}{6},n=20=2870$
Period of $\tan ky=\frac{\pi }{2870}$  
Period of  $\sin ky=\frac{2\pi }{2870}=\frac{\pi }{1435}$
 $\therefore$ Required period  $=\frac{\pi }{1435}$