Gujarati
8. Sequences and Series
easy

यदि $a$ तथा $b$ के मध्य गुणोत्तर माध्य $\frac{{{a^{n + 1}} + {b^{n +1}}}}{{{a^n} + {b^n}}}$ है, तब $n$ का मान होगा 

A

$1$

B

$-1/2$

C

$1/2$

D

$2$

Solution

(b) प्रश्नानुसार, $\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}} = {(ab)^{1/2}}$

$ \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 $ ${a^{n + 1/2}} – {b^{n + 1/2}} = 0$

$(\because \;a \ne b$

$\Rightarrow {a^{1/2}} \ne {b^{1/2}})$

$ \Rightarrow $ ${\left( {\frac{a}{b}} \right)^{n + 1/2}} = 1 = {\left( {\frac{a}{b}} \right)^0}$

$\Rightarrow n + \frac{1}{2} = 0 $

$\Rightarrow n =  – \frac{1}{2}$.

Standard 11
Mathematics

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