Question

CALCULAȚI ACESTE 2 ECUAȚII.

REZOLVARE COMPLETĂ!

Vă rog să mă ajutați la aceste exerciții de algebră.

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Answer 1

Explicație pas cu pas:

$b = \sqrt{ \dfrac{1}{ \not2} \cdot \dfrac{\not2}{\not3} \cdot \dfrac{\not3}{\not4} \cdot ... \cdot \dfrac{\not64}{65} \cdot \bigg(\dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{12} + ... + \dfrac{1}{4160} \bigg)} =$

$= \sqrt{ \dfrac{1}{65} \cdot \bigg(\dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} + ... + \dfrac{1}{64 \cdot 65} \bigg)}$

$= \sqrt{ \dfrac{1}{65} \cdot \bigg(\dfrac{1}{1} - \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + ... + \dfrac{1}{64} - \dfrac{1}{65} \bigg)}$

$= \sqrt{ \dfrac{1}{65} \cdot \bigg(\dfrac{1}{1} - \dfrac{1}{65} \bigg)} = \sqrt{ \dfrac{1}{65} \cdot \dfrac{65 - 1}{65}}$

$= \sqrt{ \dfrac{64}{ {65}^{2} }} = \sqrt{ \dfrac{ {8}^{2} }{ {65}^{2} }} = \bf \dfrac{8}{65}$

.

$c = \sqrt{ \bigg(1 + \dfrac{1}{1 + 2} + \dfrac{1}{1 + 2 + 3} + ... + \dfrac{1}{1 + 2 + 3 + ... + 199} \bigg) \cdot 2 \dfrac{2}{199} } =$

$= \sqrt{ \bigg(1 + \dfrac{1}{ \frac{2 \cdot 3}{2} } + \dfrac{1}{ \frac{3 \cdot 4}{2} } + ... + \dfrac{1}{ \frac{199 \cdot 200}{2} } \bigg) \cdot \dfrac{2 \cdot 199 + 2}{199} }$

$= \sqrt{ \bigg(1 + \dfrac{2}{2 \cdot 3} + \dfrac{2}{3 \cdot 4} + ... + \dfrac{2}{199 \cdot 200} \bigg) \cdot \dfrac{400}{199} }$

$= \sqrt{ \bigg[1 + 2 \cdot \bigg(\dfrac{1}{2} - \dfrac{1}{3}\bigg) + 2 \cdot \bigg(\dfrac{1}{3} - \dfrac{1}{4}\bigg) + ... + 2 \cdot \bigg(\dfrac{1}{199} - \dfrac{1}{200}\bigg) \bigg] \cdot \dfrac{400}{199} }$

$= \sqrt{ \bigg[1 + 2 \cdot \bigg(\dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + ... + \dfrac{1}{199} - \dfrac{1}{200} \bigg] \cdot \dfrac{400}{199} }$

$= \sqrt{ \bigg[1 + 2 \cdot \bigg(\dfrac{1}{2} - \dfrac{1}{200} \bigg] \cdot \dfrac{400}{199} }$

$= \sqrt{ \bigg(1 + 2 \cdot \dfrac{100 - 1}{200}\bigg) \cdot \dfrac{400}{199} } = \sqrt{ \bigg(1 + \dfrac{99}{100}\bigg) \cdot \dfrac{400}{199} }$

$= \sqrt{\dfrac{100 + 99}{100} \cdot \dfrac{400}{199} } = \sqrt{\dfrac{199}{100} \cdot \dfrac{400}{199} }$

$= \sqrt{4} = \bf 2$