what-is-larger-2022-2022-or-2021-2023-

Question Number 183660 by mr W last updated on 28/Dec/22

w h a t i s l a r g e r,

2022^{2022} o r 2021^{2023} ?

Answered by Ar Brandon last updated on 28/Dec/22

1^{1} > 0^{2}

2^{2} > 1^{3}

3^{3} > 2^{4}

4^{4} > 3^{5}

5^{5} < 4^{6}

6^{6} < 5^{7}

x^{x} < {(x - 1)}^{x + 1} \forall x \in N \land x ⩾ 5

\Rightarrow 2022^{2022} < 2021^{2023}

Answered by manxsol last updated on 29/Dec/22

2022^{2022} a n d 2021 \times 2021^{2022}

\frac{2022^{2022}}{2021^{2022}} a n d 2021

{(1 + \frac{1}{2021})}^{2022} = {(1 + \frac{1}{2021})}^{2021} \times (1 + \frac{1}{2021})

= e (1 + \frac{1}{2021}) = e + \frac{e}{2021}

e + \frac{e}{2021} < 2021

2022^{2022} ⟨ 2021^{2023}

t h e o r y

l i \underset{x \to \infty}{m} {(1 + \frac{1}{x})}^{x} = e = 2.71 \dots

Answered by mr W last updated on 28/Dec/22

A = 2022^{2022}

B = 2021^{2023}

\frac{A}{B} = {(\frac{2022}{2021})}^{2021} \times (\frac{2022}{2021}) \times \frac{1}{2021}

= {(1 + \frac{1}{2021})}^{2021} \times (\frac{2022}{2021}) \times \frac{1}{2021}

< e \times (\frac{2022}{2021}) \times \frac{1}{2021}

< \frac{2022 \times 3}{2021 \times 2021} < 1

\Rightarrow A < B

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