Question

$if \: \frac{ {a}^{n + 1} + {a}^{n + 1} }{ {a}^{n} + {b}^{n} } is \: the \: arithmetic \\ \: mean \: between \: \\ a \: and \: b. \: then \: find \: the \: value \: of \: n$
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Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm \: \dfrac{ {a}^{n + 1} + {a}^{n + 1} }{ {a}^{n} + {b}^{n} } is \: the \: arithmetic\: mean \: between \:a \: and \: b$

$\rm :\implies\:\: \dfrac{ {a}^{n + 1} + {a}^{n + 1} }{ {a}^{n} + {b}^{n} } \: = \: \dfrac{a + b}{2}$

$\rm :\longmapsto\:2( {a}^{n + 1} + {b}^{n + 1}) = (a + b)( {a}^{n} + {b}^{n})$

$\rm :\longmapsto\:2{a}^{n + 1} + 2{b}^{n + 1}= {a}^{n + 1} + b {a}^{n} + {b}^{n + 1} + {ab}^{n}$

$\rm :\longmapsto\:2{a}^{n + 1} + 2{b}^{n + 1} - {a}^{n + 1} - {b}^{n + 1} = b {a}^{n} + {ab}^{n}$

$\rm :\longmapsto\:{a}^{n + 1} + {b}^{n + 1} = b {a}^{n} + {ab}^{n}$

$\rm :\longmapsto\:{a}^{n + 1} - {ba}^{n } = a {b}^{n} - {b}^{n + 1}$

$\rm :\longmapsto\:{a}^{n}(a - {b})= {b}^{n}(a - {b})$

$\rm :\longmapsto\: {a}^{n} = {b}^{n}$

$\rm :\longmapsto\: {\bigg(\dfrac{a}{b} \bigg) }^{n} = 1$

$\rm :\longmapsto\: {\bigg(\dfrac{a}{b} \bigg) }^{n} = {\bigg(\dfrac{a}{b} \bigg) }^{0}$

$\bf\implies \:n = 0$

Additional Information :-

1. Arithmetic mean > Geometric mean if observations are distinct.

2. Geometric mean between 2 positive numbers a and b is

$\sf \: \sqrt{ab}$

3. If n A.M's are inserted between two numbers a and b, then sum of n A.M's is n times the single AM between a and b.