Question

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Answer 1

Explicação passo-a-passo:

1) Seja S o valor dessa expressão

Veja que:

$\sf \left(1-\dfrac{1}{2}\right)=\dfrac{1}{2}$

$\sf \left(1-\dfrac{1}{3}\right)=\dfrac{2}{3}$

$\sf \left(1-\dfrac{1}{4}\right)=\dfrac{3}{4}$

...

$\sf \left(1-\dfrac{1}{2020}\right)=\dfrac{2019}{2020}$

Assim:

$\sf \left(1-\dfrac{1}{2}\right)\cdot\left(1-\dfrac{1}{3}\right)\cdot\left(1-\dfrac{1}{4}\right)\dots\left(1-\dfrac{1}{2020}\right)$

$\sf =\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot\dfrac{3}{4}\dots\dfrac{2017}{2018}\cdot\dfrac{2018}{2019}\cdot\dfrac{2019}{2020}$

$\sf =\dfrac{1}{2020}$

Logo:

$\sf S=\dfrac{1}{\frac{1}{2020}}$

$\sf S=2020$

Letra A

2)

Temos que:

$\sf \dfrac{1}{1\cdot2}=1-\dfrac{1}{2}$

$\sf \dfrac{1}{2\cdot3}=\dfrac{1}{2}-\dfrac{1}{3}$

$\sf \dfrac{1}{3\cdot4}=\dfrac{1}{3}-\dfrac{1}{4}$

...

$\sf \dfrac{1}{2019\cdot2020}=\dfrac{1}{2019}-\dfrac{1}{2020}$

Logo:

$\sf S=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dots+\dfrac{1}{2018}-\dfrac{1}{2019}+\dfrac{1}{2019}-\dfrac{1}{2020}$

$\sf S=1-\dfrac{1}{2020}$

$\sf S=\dfrac{2020-1}{2020}$

$\sf S=\dfrac{2019}{2020}$

Letra A