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Question Number 119054 by mathocean1 last updated on 21/Oct/20

$$Showbyrecurencethat:$$$$5^n⩾1+4n;n\in \mathbb{N}$$

Answered by 1549442205PVT last updated on 22/Oct/20

$$\bullet \mathrm{For}\mathrm{n}=0\mathrm{we}\mathrm{have}{5}^0=1=1+4.0\Rightarrow \mathrm{the}$$$$\mathrm{inequality}\mathrm{is}\mathrm{true}$$$$\bullet \mathrm{Suppose}\mathrm{the}\mathrm{inequality}\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{n}=\mathrm{k}$$$$\mathrm{i}.\mathrm{e}{5}^{\mathrm{k}}⩾1+4\mathrm{k}.\mathrm{Then}{5}^{\mathrm{k}+1}=5.5^{\mathrm{k}}⩾5(1+4\mathrm{k})$$$$=5+20\mathrm{k}⩾5+4\mathrm{k}=1+4(\mathrm{k}+1)\mathrm{which}$$$$\mathrm{shows}\mathrm{the}\mathrm{inequality}\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{n}=\mathrm{k}+1$$$$\mathrm{By}\mathrm{induction}\mathrm{principle}\mathrm{the}\mathrm{inequality}\mathrm{is}$$$$\mathrm{true}\mathrm{for}\mathrm{\forall }\mathrm{n}\in \mathrm{N}(\mathrm{q}.\mathrm{e}.\mathrm{d})$$

Answered by Bird last updated on 21/Oct/20

$$n=0weget1⩾1+0\left(true\right)$$$$letsuppose{5}^n⩾1+4nandprove$$$$5^{n+1}⩾1+4(n+1)wehsve$$$$5^n⩾1+4n\left(hypothese\right)\Rightarrow$$$$5^{n+1}⩾5(1+4n)=5+20n$$$$wehave5+20n-(1+4(n+1))$$$$=5+20n-5-4n=16n⩾0\Rightarrow$$$$5^{n+1}⩾1+4(n+1)sotherelation$$$$istrueattermn+1$$

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