Question

PLEASE ANSWER FAST WITH FULL EXPLANATION.​

Answer 1

Answer:

n = -1/2

Step-by-step explanation:

Given that $\sf \dfrac{ {a}^{n + 1} + {b}^{n + 1} }{ {a}^{n} + {b}^{n} }$ is the G.M. between a and b.

Geometric mean between two number x and y is given by $\sf \sqrt{ab}$.

Therefore,

$\sf \implies\dfrac{ {a}^{n + 1} + {b}^{n + 1} }{ {a}^{n} + {b}^{n} } = \sqrt{ab}$

$\sf \implies{a}^{n + 1} + {b}^{n + 1} = \sqrt{ab} ( {a}^{n} + {b}^{n} )$

$\sf \implies{a}^{n + 1} + {b}^{n + 1} = ({a}^{n} \sqrt{ab}) + ( {b}^{n} \sqrt{ab} )$

$\sf \implies{a}^{n + 1} - ({a}^{n} \sqrt{ab}) = ( {b}^{n} \sqrt{ab} ) - {b}^{n + 1}$

$\displaystyle \sf \implies{a}^{n + 1} - ({a}^{n} {a}^{ \frac{1}{2}} {b}^{ \frac{1}{2} } ) = ( {b}^{n} {a}^{ \frac{1}{2}} {b}^{ \frac{1}{2}} ) - {b}^{n + 1}$

$\displaystyle \sf \implies{a}^{n + 1} - ({a}^{n + \frac{1}{2} } {b}^{ \frac{1}{2} } ) = ( {b}^{n + \frac{1}{2} } {a}^{ \frac{1}{2}} ) - {b}^{n + 1}$

$\displaystyle \sf \implies{a}^{n + \frac{1}{2} } ( {a}^{ \frac{1}{2} } -{b}^{ \frac{1}{2} } ) = {b}^{n + \frac{1}{2} } ( {a}^{ \frac{1}{2} } -{b}^{ \frac{1}{2} } )$

$\displaystyle \sf \implies{a}^{n + \frac{1}{2} } = {b}^{n + \frac{1}{2} }$

$\displaystyle \sf \implies \frac{{a}^{n + \frac{1}{2} }}{{b}^{n + \frac{1}{2} } } = 1$

$\displaystyle \sf \implies \left(\frac{a}{b}\right)^{n + \frac{1}{2}}= 1$

$\displaystyle \sf \implies \left(\frac{a}{b}\right)^{n + \frac{1}{2}}= \left(\frac{a}{b} \right)^0$

$\sf \implies n + \dfrac{1}{2} = 0$

$\boxed{\sf \implies n = - \dfrac{1}{2} }$

This is the required value of n.

Answer 2

Answer:

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