Question

Find the value of n if a^n+1 + b^n+1 / a^n + b^n is the G.M

Answer 1

Heya !
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★Geometric Mean ( G.M ) of two numbers a and b is given by

$= > \sqrt{ab}$

→ So According to the Question we have ,

$= > \frac{a {}^{n + 1} + b {}^{n + 1} }{a {}^{n} + b {}^{n} } = \sqrt{ab} \\ \\ = > squaring \: both \: the \: sides \: we \: have \: \\ \\ = > \frac{(a {}^{n + 1 } + b {}^{n + 1} ) {}^{2} }{(a {}^{n} + b {}^{n} ) {}^{2} } = ab$

$= > a {}^{2n + 2} + b {}^{2n + 2} + 2 \times a {}^{n + 1} \times b {}^{n + 1} = ab(a {}^{2n} + b {}^{2n} + 2 \times a {}^{n} \times b {}^{n} )$

$= > a {}^{2n + 2} + b {}^{2n + 2} = a {}^{2n + 1} b + ab {}^{2n + 1} \\ \\ = > a {}^{2n + 2} - a {}^{2n + 1} b = ab {}^{2n + 1} - b {}^{2n + 2} \\ \\ = > a {}^{2n + 1} (a - b) = b {}^{2n + 1} (a - b)$

★Now ( a - b ) gets canceled out!!

$= > ( \frac{a}{b} ) {}^{2n + 1}$

•°• We have ,

→ 2n + 1 = 0


Thus, n = - 1/2
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Answer 2

Given that

a^n+1 + b^n+1 /a^n + b^n is geometric mean..

G.M = √ab

So we will get

a^n+1 + b^n+1 /a^n + b^n = √ab

square &Cross multiply

We have , (a/b)2n+1

Means 2n+1 = 0

So n = -1/2