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Question Number 142085 by Rexzie last updated on 30/May/21

$$Proofthat1+3n<n^{2}foreverypositiveintegern⩾4$$

Answered by MJS_new last updated on 26/May/21

$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{wrong}\mathrm{for}\frac{3-\sqrt{13}}{2}⩽n⩽\frac{3+\sqrt{13}}{2}$$

Answered by physicstutes last updated on 26/May/21

$$\mathrm{Thesame}\mathrm{as}\mathrm{proving}\mathrm{for},$$ $$0<n^2-3n-1$$ $$\mathrm{or}{n}^2-3n-1>0$$ $${(n-\frac{3}{2})}^2-\frac{9}{4}-1>0$$ $${(n-\frac{3}{2})}^2-\frac{13}{4}>0$$ $$\mathrm{\forall }n\in \mathbb{R},{(n-\frac{3}{2})}^2⩾0,\mathrm{but}\mathrm{\forall }n\in \mathbb{R},\nRightarrow {(n-\frac{3}{2})}^2-\frac{13}{4}>0$$ $$\mathrm{take}\mathrm{the}\mathrm{case}\mathrm{of}\mathrm{the}\mathrm{the}\mathrm{interval}\mathrm{posted}\mathrm{above}.$$

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