Find-n-so-that-a-n-1-b-n-1-a-n-b-n-may-be-the-arithmetic-mean-between-a-and-b-

Question Number 160058 by Rasheed.Sindhi last updated on 24/Nov/21

F i n d n s o t h a t \frac{a^{n + 1} + b^{n + 1}}{a^{n} + b^{n}} m a y b e

t h e a r i t h m e t i c m e a n b e t w e e n a

a n d b .

Commented by Tinku Tara last updated on 24/Nov/21

n = 0

Commented by Rasheed.Sindhi last updated on 24/Nov/21

R i g h t s i r, p r o c e s s p l e a s e .

Answered by Kunal12588 last updated on 24/Nov/21

\frac{a + b}{2} = \frac{a^{n + 1} + b^{n + 1}}{a^{n} + b^{n}}

\Rightarrow a^{n + 1} + {a b}^{n} + a^{n} b + b^{n + 1} = 2 a^{n + 1} + 2 b^{n + 1}

\Rightarrow a^{n} a - {a b}^{n} - a^{n} b + b^{n} b = 0

\Rightarrow a (a^{n} - b^{n}) - b (a^{n} - b^{n}) = 0

\Rightarrow (a - b) (a^{n} - b^{n}) = 0

\Rightarrow a = b o r a^{n} = b^{n}

a^{n} = b^{n} \Rightarrow {(\frac{a}{b})}^{n} = 1 \Rightarrow n = 0

Commented by Rasheed.Sindhi last updated on 24/Nov/21

T h a n X s i r!