Explanation

(b) Time period of spring
$T = 2\pi\sqrt{\left( \frac{m}{k} \right)}$
$k$, being the force constant of spring
For first spring
$t_{1} = 2\pi\sqrt{\left( \frac{m}{k_{1}} \right)}$ …(i)
For second spring
$t_{2} = 2\pi\sqrt{\left( \frac{m}{k_{2}} \right)}$ …(ii)
The effective force constant in their series combination is
$k = \frac{k_{1}k_{2}}{k_{1} + k_{2}}$
$\therefore$ Time period of combination
$T = 2\pi\sqrt{\left( \frac{m(k_{1} + k_{2})}{k_{1}k_{2}} \right)}$
$\Rightarrow$ $T^{2} = \frac{4\pi^{2}m(k_{1} + k_{2})}{k_{1}k_{2}}$ …(iii)
Form Eqs. (i) and (ii), we obtain
$t_{1}^{2} + t_{2}^{2} = 4\pi^{2}\left( \frac{m}{k_{1}} + \frac{m}{k_{2}} \right)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ t_{1}^{2} + t_{2}^{2} = 4\pi^{2}m\left( \frac{1}{k_{1}} + \frac{1}{k_{2}} \right)$
$\Rightarrow$ $t_{1}^{2} + t_{2}^{2} = \frac{4\pi^{2}m\left( k_{1} + k_{2} \right)}{k_{1}k_{2}}$
$\therefore$ $t_{1}^{2} + t_{2}^{2} = T^{2}$ [from Eq. (iii)]