Đáp án: So sánh √2021 -√2020 và √2022 -√2021 - Đặt √(2021) - √(2020) = a √(2022) - √(2021) = b ⇒ a - b = √(2021) - √(2020) - (√(2022) - √(2021)) = √(2021) - √(2020) - √(2022) + √(2021)) = 2√(2021) - (√(2020) + √(2022)) Ta lại có • (√(2020) + √(2022))^2 = 2020 + 2022 + 2√(20202022) = 4042 + 2√((2021 - 1)(2021 + 1)) = 4042 + 2√(2021^2 - 1) • (2√(2...

Đặt √(2021) - √(2020) = a √(2022) - √(2021) = b ⇒ a - b = √(2021) - √(2020) - (√(2022) - √(2021)) = √(2021) - √(2020) - √(2022) + √(2021)) = 2√(2021) - (√(2020) + √(2022)) Ta lại có • (√(2020) + √(2022))^2 = 2020 + 2022 + 2√(20202022) = 4042 + 2√((2021 - 1)(2021 + 1)) = 4042 + 2√(2021^2 - 1) • (2√(2...

09062007

6 tháng trước

So sánh √2021 -√2020 và √2022 -√2021

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ThuUyen2107

6 tháng trước

Đặt: $\sqrt{2021} - \sqrt{2020} = a$

$\sqrt{2022} - \sqrt{2021} = b$

$⇒ a - b = \sqrt{2021} - \sqrt{2020} - (\sqrt{2022} - \sqrt{2021})$

$= \sqrt{2021} - \sqrt{2020} - \sqrt{2022} + \sqrt{2021})$

$= 2\sqrt{2021} - (\sqrt{2020} + \sqrt{2022})$

$\\$

Ta lại có:

$\bullet$ $(\sqrt{2020} + \sqrt{2022})^2$

$= 2020 + 2022 + 2\sqrt{2020.2022}$

$= 4042 + 2\sqrt{(2021 - 1).(2021 + 1)}$

$= 4042 + 2\sqrt{2021^2 - 1}$

$\bullet$ $(2\sqrt{2021})^2 = 4.2021 = 4042 + 2.2021 = 4042 + 2\sqrt{2021^2}$

Vì: $2\sqrt{2021^2 - 1} < 2\sqrt{2021^2}$

$⇒ 4042 + 2\sqrt{2021^2 - 1} < 4042 + 2\sqrt{2021^2}$

Hay: $(\sqrt{2020} + \sqrt{2022})^2 < (2\sqrt{2021})^2$

$⇒ \sqrt{2020} + \sqrt{2022} < 2\sqrt{2021}$

$⇔ 2\sqrt{2021} - (\sqrt{2020} + \sqrt{2022}) > 0$

$⇒ a - b > 0$

$⇔ a > b$

Vậy: $\sqrt{2021} - \sqrt{2020} > \sqrt{2022} - \sqrt{2021}$

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