Who-is-greater-70-71-or-71-70-

Question Number 183535 by HeferH last updated on 26/Dec/22

W h o i s g r e a t e r ? 70^{71} o r 71^{70}

Answered by Frix last updated on 26/Dec/22

1^{2} < 2^{1}

2^{3} < 3^{2}

===

3^{4} > 4^{3}

4^{5} > 5^{4}

\dots

n^{n + 1} > {(n + 1)}^{n} \forall n ⩾ 3

Commented by HeferH last updated on 26/Dec/22

t h a n k s :)

Answered by mr W last updated on 26/Dec/22

f (x) = x^{\frac{1}{x}} = e^{\frac{\ln x}{x}}

f^{'} (x) = x^{\frac{1}{x}} (\frac{1 - \ln x}{x^{2}})

w e s e e f o r x > e, f^{'} (x) < 0, t h a t m e a n s

f (x) i s s t r i c t l y d e c r e a s i n g f o r x > e,

i . e . 3^{\frac{1}{3}} > 4^{\frac{1}{4}} > 5^{\frac{1}{5}} > \dots

70^{\frac{1}{70}} > 71^{\frac{1}{71}}

70^{\frac{71}{70}} > 71

\Rightarrow 70^{71} > 71^{70}

s i m i l a r l y y o u c a n a l s o g e t e . g .

100^{200} > 200^{100}

Commented by HeferH last updated on 26/Dec/22

t h a n k s!

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