Question

Обчисліть::::::::::::::::::::::::

Answer 1

Ответ:

2021

Объяснение:

$1 - \displaystyle\frac{1}{{{{20}^2}}} = \displaystyle\frac{{{{20}^2} - 1}}{{{{20}^2}}} = \displaystyle\frac{{(20 - 1)(20 + 1)}}{{{{20}^2}}} = \displaystyle\frac{{19 \cdot 21}}{{{{20}^2}}};$

$1 - \displaystyle\frac{1}{{{{21}^2}}} = \displaystyle\frac{{{{21}^2} - 1}}{{{{21}^2}}} = \displaystyle\frac{{(21 - 1)(21 + 1)}}{{{{21}^2}}} = \displaystyle\frac{{20 \cdot 22}}{{{{21}^2}}};$

$1 - \displaystyle\frac{1}{{{{22}^2}}} = \displaystyle\frac{{{{22}^2} - 1}}{{{{22}^2}}} = \displaystyle\frac{{(22 - 1)(22 + 1)}}{{{{22}^2}}} = \displaystyle\frac{{21 \cdot 23}}{{{{22}^2}}};$

$\left( {1 - \displaystyle\frac{1}{{{{20}^2}}}} \right)\left( {1 - \displaystyle\frac{1}{{{{21}^2}}}} \right)\left( {1 - \displaystyle\frac{1}{{{{22}^2}}}} \right) = \displaystyle\frac{{19 \cdot 21}}{{{{20}^2}}} \cdot \displaystyle\frac{{20 \cdot 22}}{{{{21}^2}}} \cdot \displaystyle\frac{{21 \cdot 23}}{{{{22}^2}}} = \displaystyle\frac{{19 \cdot 23}}{{20 \cdot 22}};$

$\displaystyle\frac{{19 \cdot 23}}{{20 \cdot 22}} + \displaystyle\frac{3}{{{{21}^2} - 1}} = \displaystyle\frac{{19 \cdot 23}}{{20 \cdot 22}} + \displaystyle\frac{3}{{(21 - 1)(21 + 1)}} = \displaystyle\frac{{19 \cdot 23 + 3}}{{20 \cdot 22}} = \displaystyle\frac{{440}}{{20 \cdot 22}} = 1;\\\\1 + 2020 = 2021.$