If-the-arithmetic-mean-of-a-and-b-is-a-n-1-b-n-1-2-show-that-n-0-

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