Question Number 178240 by Shrinava last updated on 14/Oct/22 | ||
$$\frac{\mathrm{a}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} }{\mathrm{a}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{n}}} }\:=\:\sqrt{\mathrm{ab}}\:\:\:\mathrm{find}\:\:\:\mathrm{n}=? \\ $$ | ||
Answered by Rasheed.Sindhi last updated on 14/Oct/22 | ||
$$\mathrm{a}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} =\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{2}}} \mathrm{b}^{\frac{\mathrm{1}}{\mathrm{2}}} \:\left(\mathrm{a}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{n}}} \right) \\ $$$$\mathrm{a}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} =\mathrm{a}^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} \mathrm{b}^{\frac{\mathrm{1}}{\mathrm{2}}} \:+\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{2}}} \mathrm{b}^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\mathrm{a}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} −\mathrm{a}^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} \mathrm{b}^{\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{2}}} \mathrm{b}^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} −\mathrm{b}^{\mathrm{n}+\mathrm{1}} \\ $$$$\mathrm{a}^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} \overset{×} {\left(\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{2}}} −\mathrm{b}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)}=\mathrm{b}^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} \overset{×} {\left(\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{2}}} −\mathrm{b}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)}\:\left[\mathrm{a}\neq\mathrm{b}\right] \\ $$$$\mathrm{a}^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{b}^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{\boldsymbol{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{1}=\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{\mathrm{0}} \\ $$$$\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0} \\ $$$$\mathrm{n}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$ | ||
Commented by mr W last updated on 14/Oct/22 | ||
$${nice}\:{solution}! \\ $$ | ||
Commented by Rasheed.Sindhi last updated on 14/Oct/22 | ||
$$\mathcal{T}{hanks}\:\boldsymbol{{sir}}! \\ $$ | ||
Commented by Tawa11 last updated on 14/Oct/22 | ||
$$\mathrm{Great}\:\mathrm{sir} \\ $$ | ||