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Question Number 175211 by MathsFan last updated on 23/Aug/22

Assume that the sequence terms tend to the constant value u, so that as n→∞, u_(n−1) →u and u_n →u. (i) show that u^2 +u−1=0 (ii) show that (1/(1+(1/(1+(1/(1+(1/(1+.....))))))))=((−1+(√5))/2)

$$\boldsymbol{\mathrm{Assume}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sequence}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{tend}} \\ $$$$\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{constant}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{{u}},\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{as}} \\ $$$$\boldsymbol{\mathrm{n}}\rightarrow\infty,\:\boldsymbol{{u}}_{\boldsymbol{{n}}−\mathrm{1}} \rightarrow\boldsymbol{{u}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{u}}_{\boldsymbol{{n}}} \rightarrow\boldsymbol{{u}}. \\ $$$$\:\left(\boldsymbol{\mathrm{i}}\right)\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\:\boldsymbol{{u}}^{\mathrm{2}} +\boldsymbol{{u}}−\mathrm{1}=\mathrm{0} \\ $$$$\:\left(\boldsymbol{\mathrm{ii}}\right)\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\:\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+.....}}}}=\frac{−\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$

Answered by Rasheed.Sindhi last updated on 23/Aug/22

(ii) u=(1/(1+(1/(1+(1/(1+(1/(1+.....)))))))); u>0 u=(1/(1+u)) u^2 +u−1=0 u=((−1+(√(1−4(1)(−1))))/2)=((−1+(√5))/2) [((−1−(√5))/2)<0]

$$\left(\boldsymbol{\mathrm{ii}}\right)\:\boldsymbol{\mathrm{u}}=\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+.....}}}};\:\boldsymbol{\mathrm{u}}>\mathrm{0} \\ $$$$\:\:\:\boldsymbol{\mathrm{u}}=\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{\mathrm{u}}} \\ $$$$\boldsymbol{\mathrm{u}}^{\mathrm{2}} +\boldsymbol{\mathrm{u}}−\mathrm{1}=\mathrm{0} \\ $$$$\boldsymbol{\mathrm{u}}=\frac{−\mathrm{1}+\sqrt{\mathrm{1}−\mathrm{4}\left(\mathrm{1}\right)\left(−\mathrm{1}\right)}}{\mathrm{2}}=\frac{−\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\left[\frac{−\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}<\mathrm{0}\right] \\ $$

Commented by MathsFan last updated on 24/Aug/22

thank you sir

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

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