A given object takes n times more time to slide down a 45 ^ o rough...

Time taken to slide along a smooth surface is ${t}_{1}$.
$S = \dfrac{1}{2}gsin {45}^{o}{t}_{1}^{2}$
${t}_{1} = \sqrt{\dfrac{2\sqrt{2s}}{g}}$
Time taken to slide along a rough surface is ${t}_{2}$.
$S = \dfrac{1}{2}\left(gsin {45}^{o} - \mu gsin {45}^{o}\right){t}_{2}^{2}$
${t}_{2} = \sqrt{\dfrac{2\sqrt{2s}}{g\left(1 - \mu\right)}}$
${t}_{2} = n{t}_{1}$
$\dfrac{2\sqrt{2s}}{g\left(1 - \mu\right)} = {n}^{2} \times \dfrac{2\sqrt{2s}}{g} ⇒ 1 - \mu = \dfrac{1}{{n}^{2}} ⇒ \mu = 1 - \dfrac{1}{{n}^{2}}$