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Question

# $\mathrm{Let}\mathrm{P}\left(n\right)\mathrm{be}\mathrm{the}\mathrm{statement}:{2}^{n}\ge 3n.\mathrm{If}\mathrm{P}\left(r\right)\mathrm{is}\mathrm{true},\mathrm{then}\mathrm{show}\mathrm{that}\mathrm{P}\left(r+1\right)\mathrm{is}\mathrm{true}.\mathrm{Do}\mathrm{you}\mathrm{conclude}\mathrm{that}\mathrm{P}\left(n\right)\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{all}n\in \mathbf{N}?$

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Solution

## $\mathrm{Since},\mathrm{for}n=1\mathrm{i}.\mathrm{e}.\mathrm{P}\left(1\right):\phantom{\rule{0ex}{0ex}}\mathrm{LHS}={2}^{1}=2\phantom{\rule{0ex}{0ex}}\mathrm{RHS}=3×1=3\phantom{\rule{0ex}{0ex}}\mathrm{As},\mathrm{LHS}<\mathrm{RHS}\phantom{\rule{0ex}{0ex}}\mathrm{So},\mathrm{it}\mathrm{is}\mathrm{not}\mathrm{true}\mathrm{for}n=1.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{we}\mathrm{conclude}\mathrm{that}\mathrm{P}\left(n\right)\mathrm{is}\mathrm{not}\mathrm{true}\mathrm{for}\mathrm{all}n\in \mathbf{N}.$

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