Question

calculate the no. of molecule of sulphur (S8) present in 24 g of solid sulphur​

Answer 1

Explanation:

n=k ,

$\begin{gathered} \\ \tt P(k) \underset{(k \: \text{ radical \: sign})}{=\sqrt{2+\sqrt{2+\sqrt{2+\ldots+\ldots+\sqrt{2}}}}=2 \cos \left(\frac{\pi}{2^{k+1}}\right) }\end{gathered}$

For n=k+1

$\begin{gathered} \begin{aligned} \tt P(k+1) & \tt \underset{ k + 1\text { radical sign }}{=\sqrt{2+\sqrt{2+\sqrt{2+\ldots+\ldots+\sqrt{2}}}}} \\ \\ & \tt=\sqrt{\{2+P(k)\}} \\ \\ & \tt=\sqrt{2+2 \cos \left(\frac{\pi}{2^{k+1}}\right)} \text { (By assumption step) } \\ \\ & \tt=\sqrt{2\left(1+\cos \left(\frac{\pi}{2^{k+1}}\right)\right)} \\ \\ & \tt=\sqrt{2\left(1+2 \cos ^{2}\left(\frac{\pi}{2^{k+1}}\right)-1\right)} \\ \\ & \tt=\sqrt{4 \cos ^{2}\left(\frac{\pi}{2^{k+2}}\right)} \\ \\ & \tt=2 \cos \left(\frac{\pi}{2^{k+2}}\right) \end{aligned} \end{gathered}$

This shows that the result is true for n=k+1. Hence by the principle of mathematical induction, the result is true for all n ∈ N .