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Question Number 56772 by Hassen_Timol last updated on 23/Mar/19

Given that :      (1/( Φ ))  =  (Φ/( 1 + Φ ))    Prove that : without using the exact                             value of Φ...       (( 1 )/( Φ ))  =  Φ − 1      Thank you

$$\mathrm{Given}\:\mathrm{that}\:: \\ $$$$\:\:\:\:\frac{\mathrm{1}}{\:\Phi\:}\:\:=\:\:\frac{\Phi}{\:\mathrm{1}\:+\:\Phi\:} \\ $$$$ \\ $$$$\mathrm{Prove}\:\mathrm{that}\::\:{without}\:{using}\:{the}\:{exact} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{value}\:{of}\:\Phi... \\ $$$$\:\:\:\:\:\frac{\:\mathrm{1}\:}{\:\Phi\:}\:\:=\:\:\Phi\:−\:\mathrm{1} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you} \\ $$

Answered by JDamian last updated on 23/Mar/19

  (1/( Φ ))  =  (Φ/( 1 + Φ ))  1+Φ=Φ^2   1=Φ^2 −Φ  1=Φ(Φ−1)  (1/Φ)=Φ−1

$$ \\ $$$$\frac{\mathrm{1}}{\:\Phi\:}\:\:=\:\:\frac{\Phi}{\:\mathrm{1}\:+\:\Phi\:} \\ $$$$\mathrm{1}+\Phi=\Phi^{\mathrm{2}} \\ $$$$\mathrm{1}=\Phi^{\mathrm{2}} −\Phi \\ $$$$\mathrm{1}=\Phi\left(\Phi−\mathrm{1}\right) \\ $$$$\frac{\mathrm{1}}{\Phi}=\Phi−\mathrm{1} \\ $$

Commented by Hassen_Timol last updated on 23/Mar/19

Thank you sir

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

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