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Question Number 85762 by Power last updated on 24/Mar/20

Commented by MJS last updated on 24/Mar/20

$$\approx 1.43312742672$$

Commented by I want to learn more last updated on 24/Mar/20

$$\mathbf{If}\mathbf{x}=\lceil 1,2,3,4,5\rceil$$$$\therefore \mathbf{x}=1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5+\mathbf{x}}}}}$$$$\therefore \mathbf{x}=1+\frac{1}{2+\frac{1}{3+\frac{5+\mathbf{x}}{21+4\mathbf{x}}}}$$$$\therefore \mathbf{x}=1+\frac{1}{2+\frac{1}{\frac{63+12\mathbf{x}+5+\mathbf{x}}{21+4\mathbf{x}}}}$$$$\therefore \mathbf{x}=1+\frac{1}{2+\frac{21+4\mathbf{x}}{68+13\mathbf{x}}}$$$$\therefore \mathbf{x}=1+\frac{1}{\frac{136+26\mathbf{x}+21+4\mathbf{x}}{68+13\mathbf{x}}}$$$$\therefore \mathbf{x}=1+\frac{68+13\mathbf{x}}{157+30\mathbf{x}}$$$$\therefore \mathbf{x}=\frac{157+30\mathbf{x}+68+13\mathbf{x}}{157+30\mathbf{x}}$$$$\therefore \mathbf{x}=\frac{225+43\mathbf{x}}{157+30\mathbf{x}}$$$$\therefore 30{\mathbf{x}}^2+157\mathbf{x}=225+43\mathbf{x}$$$$\therefore 30{\mathbf{x}}^2+114\mathbf{x}-225=0$$$$\mathbf{a}=30,\mathbf{b}=114,\mathbf{c}=-225$$$$\therefore \mathbf{x}=\approx 1.433166662$$

Commented by Power last updated on 24/Mar/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Commented by MJS last updated on 24/Mar/20

$$\mathrm{but}\mathrm{the}\mathrm{question}\mathrm{includes}‘‘..{.}^{″},\mathrm{which}\mathrm{means}$$$$\mathrm{to}\mathrm{infinity}.\mathrm{you}\mathrm{simply}\mathrm{ignored}\mathrm{this}.$$

Commented by I want to learn more last updated on 24/Mar/20

$$\mathrm{It}\mathrm{is}\mathrm{a}\mathrm{recurring}\lceil 1,2,3,4,5\rceil \mathrm{sir}$$

Commented by I want to learn more last updated on 24/Mar/20

$$\mathrm{I}\mathrm{think}\mathrm{it}\mathrm{approximate}\mathrm{to}\mathrm{your}\mathrm{answer}\mathrm{sir}$$

Commented by MJS last updated on 24/Mar/20

$$\mathrm{I}\mathrm{think}{\mathrm{it}}^{\prime }\mathrm{s}[1,2,3,4,5,6,...]$$$$\mathrm{my}\mathrm{value}\mathrm{is}\mathrm{an}\mathrm{approximation}\mathrm{of}\mathrm{this}$$

Commented by I want to learn more last updated on 24/Mar/20

$$\mathrm{Oh}.\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}\mathrm{sir}.\mathrm{i}\mathrm{used}\lceil 1,2,3,4,5\rceil \mathrm{as}\mathrm{shown}\mathrm{in}\mathrm{the}\mathrm{question}.$$

Commented by MJS last updated on 24/Mar/20

$$\mathrm{if}\mathrm{I}\mathrm{give}\mathrm{you}\mathrm{a}\mathrm{sequence}{a}_n=⟨1,2,3,4,5,...⟩$$$$\mathrm{would}\mathrm{you}\mathrm{really}\mathrm{take}\mathrm{it}\mathrm{as}$$$$a_n=⟨1,2,3,4,5,1,2,3,4,5,...⟩?$$$$\mathrm{I}\mathrm{guess}\mathrm{no}$$

Commented by I want to learn more last updated on 24/Mar/20

$$\mathrm{That}\mathrm{is}\mathrm{infinite}\mathrm{continue}\mathrm{fraction}\mathrm{sir}.$$$$\mathrm{It}\mathrm{is}\mathrm{assume}\mathrm{that}\mathrm{the}\mathrm{sequence}\mathrm{repeat}\mathrm{in}\mathrm{that}\mathrm{manner}.$$$$\mathrm{As}\mathrm{the}\mathrm{dot}\mathrm{dot}\mathrm{dot}...$$

Commented by MJS last updated on 24/Mar/20

$$\mathrm{this}\mathrm{is}\mathrm{just}\mathrm{your}\mathrm{interpretation}$$$$x=2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+...}}}}}}}}}$$$$x=\mathrm{e}\mathrm{if}\mathrm{we}\mathrm{continue}$$$$2+[1,2,1,1,4,1,1,6,1,1,8,...]$$$$\mathrm{if}\mathrm{I}\mathrm{want}\mathrm{a}\mathrm{repetition}\mathrm{I}\mathrm{must}\mathrm{show}\mathrm{the}\mathrm{first}$$$$\mathrm{repeated}\mathrm{number}\left(\mathrm{s}\right)\mathrm{like}\mathrm{this}:$$$$2+[1,2,1,1,4,1,1,6,1,1,2,...]$$

Commented by I want to learn more last updated on 25/Mar/20

$$\mathrm{Ok}\mathrm{sir}.\mathrm{Thanks}.$$

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