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Question Number 20652 by Tinkutara last updated on 30/Aug/17

$$\mathrm{Suppose}x\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{real}\mathrm{number}$$$$\mathrm{such}\mathrm{that}\left\{x\right\},\left[x\right]\mathrm{and}x\mathrm{are}\mathrm{in}\mathrm{a}$$$$\mathrm{geometric}\mathrm{progression}.\mathrm{Find}\mathrm{the}\mathrm{least}$$$$\mathrm{positive}\mathrm{integer}n\mathrm{such}\mathrm{that}{x}^n>100.$$$$(\mathrm{Here}\left[x\right]\mathrm{denotes}\mathrm{the}\mathrm{integer}\mathrm{part}\mathrm{of}x$$$$\mathrm{and}\left\{x\right\}=x-\left[x\right].)$$

Commented by ajfour last updated on 30/Aug/17

$$Isitn>\frac{2}{\log {}_{10}\left(\frac{\sqrt{5}+1}{2}\right)}$$$$\Rightarrow {\left(n\right)}_{min}=10.$$

Commented by Tinkutara last updated on 30/Aug/17

$$\mathrm{Yes}.\mathrm{But}\mathrm{can}\mathrm{it}\mathrm{be}\mathrm{solved}\mathrm{in}\mathrm{an}\mathrm{exam}$$$$\mathrm{where}\mathrm{calculators}\mathrm{and}\log \mathrm{tables}\mathrm{are}$$$$\mathrm{not}\mathrm{allowed}?\mathrm{I}\mathrm{know}\mathrm{the}\mathrm{solution}\mathrm{so}\mathrm{no}$$$$\mathrm{need}\mathrm{to}\mathrm{post}\mathrm{it}.$$

Commented by ajfour last updated on 30/Aug/17

$$\log \left(\frac{\sqrt{5}+1}{2}\right)=\log \left(\frac{2.236+1}{2}\right)$$$$=\log (1.618)=\log (16.18)-1$$$$\approx 4(0.3010)-1\approx 0.2$$$$\frac{2}{\log \left(\frac{\sqrt{5}+1}{2}\right)}\approx 10.$$

Commented by Tinkutara last updated on 31/Aug/17

$$\mathrm{Thanks}!$$

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