If the arithmetic mean of a and b is ((a^(n+1) +b^(n+1) )/2), show that n=0


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Question Number 28320 by NECx last updated on 24/Jan/18

$I f t h e a r i t h m e t i c m e a n o f a a n d$ $b i s \frac{a^{n + 1} + b^{n + 1}}{2}, s h o w t h a t n = 0$
Commented by mrW2 last updated on 24/Jan/18

$t h i s i s n o t a l w a y s t r u e, s i r .$ $e . g . i f a = b = 1, o r i f a = 1 a n d b = 0, n$ $c a n b e a n y v a l u e .$
Commented by NECx last updated on 24/Jan/18

$t h a n k s f o r t h e c o r r e c t i o n$
Commented by abdo imad last updated on 24/Jan/18

$\frac{a^{n + 1} + b^{n + 1}}{2} = \frac{a + b}{2} \Leftrightarrow a^{n + 1} - a = b^{n + 1} - b \forall a a n d b$ $\Rightarrow \frac{\partial}{\partial a} (a^{n + 1} - a) = 0 f o r b f i x e d \Rightarrow (n + 1) a^{n} - 1 = 0 \forall a$ $\Rightarrow (n + 1) a^{n} = 1 \forall a \Rightarrow n = 0$
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