If the geometric mean between a and b is frac | vedclass

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Question

(b) As given $\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}} = {(ab)^{1/2}}$

$ \Rightarrow $ ${a^{n + 1}} - {a^{n + 1/2}}{b^{1/2}} + {b^{n + 1

Vedclass · Accepted Answer

$-1/2$

Vedclass · Answer

(b) As given $\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}} = {(ab)^{1/2}}$

$ \Rightarrow $ ${a^{n + 1}} - {a^{n + 1/2}}{b^{1/2}} + {b^{n + 1}} - {a^{1/2}}{b^{n + 1/2}} = 0$

$ \Rightarrow $ $({a^{n + 1/2}} - {b^{n + 1/2}})({a^{1/2}} - {b^{1/2}}) = 0$

$ \Rightarrow $ ${a^{n + 1/2}} - {b^{n + 1/2}} = 0$                                           

$(\because \;a 
e b \Rightarrow {a^{1/2}} 
e {b^{1/2}})$                            

$ \Rightarrow $ ${\left( {\frac{a}{b}} ight)^{n + 1/2}} = 1 = {\left( {\frac{a}{b}} ight)^0} $

$\Rightarrow n + \frac{1}{2} = 0$

$\Rightarrow n = - \frac{1}{2}$.

If the geometric mean between $a$ and $b$ is $\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}}$, then the value of $n$ is

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