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

what is larger, 2022^(2022) or 2021^(2023) ?

$${what}\:{is}\:{larger}, \\ $$$$\mathrm{2022}^{\mathrm{2022}} \:{or}\:\mathrm{2021}^{\mathrm{2023}} \:? \\ $$

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) ∀x∈N ∧ x≥5 ⇒2022^(2022) < 2021^(2023)

$$\mathrm{1}^{\mathrm{1}} >\mathrm{0}^{\mathrm{2}} \\ $$$$\mathrm{2}^{\mathrm{2}} >\mathrm{1}^{\mathrm{3}} \\ $$$$\mathrm{3}^{\mathrm{3}} >\mathrm{2}^{\mathrm{4}} \\ $$$$\mathrm{4}^{\mathrm{4}} >\mathrm{3}^{\mathrm{5}} \\ $$$$\mathrm{5}^{\mathrm{5}} <\mathrm{4}^{\mathrm{6}} \\ $$$$\mathrm{6}^{\mathrm{6}} <\mathrm{5}^{\mathrm{7}} \\ $$$${x}^{{x}} <\left({x}−\mathrm{1}\right)^{{x}+\mathrm{1}} \:\forall{x}\in\mathbb{N}\:\wedge\:{x}\geqslant\mathrm{5} \\ $$$$\Rightarrow\mathrm{2022}^{\mathrm{2022}} \:<\:\mathrm{2021}^{\mathrm{2023}} \\ $$

Answered by manxsol last updated on 29/Dec/22

2022^(2022) and 2021×2021^(2022) ((2022^(2022) )/(2021^(2022) )) and 2021 (1+(1/(2021)))^(2022) =(1+(1/(2021)))^(2021) ×(1+(1/(2021))) =e(1+(1/(2021)))=e+(e/(2021)) e+(e/(2021)) <2021 2022^(2022) ⟨ 2021^(2023) theory lim_(x→∞) (1+(1/x))^x =e=2.71...

$$\mathrm{2022}^{\mathrm{2022}} {and}\:\mathrm{2021}×\mathrm{2021}^{\mathrm{2022}} \\ $$$$\frac{\mathrm{2022}^{\mathrm{2022}} }{\mathrm{2021}^{\mathrm{2022}} }\:\:{and}\:\mathrm{2021} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2021}}\right)^{\mathrm{2022}} =\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2021}}\right)^{\mathrm{2021}} ×\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2021}}\right) \\ $$$$={e}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2021}}\right)={e}+\frac{{e}}{\mathrm{2021}} \\ $$$${e}+\frac{{e}}{\mathrm{2021}}\:<\mathrm{2021} \\ $$$$\mathrm{2022}^{\mathrm{2022}} \:\langle\:\:\mathrm{2021}^{\mathrm{2023}} \\ $$$${theory} \\ $$$${li}\underset{{x}\rightarrow\infty} {{m}}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{{x}} ={e}=\mathrm{2}.\mathrm{71}... \\ $$$$ \\ $$$$ \\ $$

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

A=2022^(2022) B=2021^(2023) (A/B)=(((2022)/(2021)))^(2021) ×(((2022)/(2021)))×(1/(2021)) =(1+(1/(2021)))^(2021) ×(((2022)/(2021)))×(1/(2021)) <e×(((2022)/(2021)))×(1/(2021)) <((2022×3)/(2021×2021))<1 ⇒A<B

$${A}=\mathrm{2022}^{\mathrm{2022}} \\ $$$${B}=\mathrm{2021}^{\mathrm{2023}} \\ $$$$\frac{{A}}{{B}}=\left(\frac{\mathrm{2022}}{\mathrm{2021}}\right)^{\mathrm{2021}} ×\left(\frac{\mathrm{2022}}{\mathrm{2021}}\right)×\frac{\mathrm{1}}{\mathrm{2021}} \\ $$$$=\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2021}}\right)^{\mathrm{2021}} ×\left(\frac{\mathrm{2022}}{\mathrm{2021}}\right)×\frac{\mathrm{1}}{\mathrm{2021}} \\ $$$$<{e}×\left(\frac{\mathrm{2022}}{\mathrm{2021}}\right)×\frac{\mathrm{1}}{\mathrm{2021}} \\ $$$$<\frac{\mathrm{2022}×\mathrm{3}}{\mathrm{2021}×\mathrm{2021}}<\mathrm{1} \\ $$$$\Rightarrow{A}<{B} \\ $$

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