Question

√2x(1+2+3+.... +50) x(1/1x2+1/2x3+1/3x4+..... +1/50x51) ​

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

Explicație pas cu pas:

$\sqrt{2 \cdot (1 + 2 + 3 + ... + 50) \cdot \Big(\dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} + ... + \dfrac{1}{50 \cdot 51}\Big)} =$

$= \sqrt{2 \cdot \dfrac{50 \cdot (50 + 1)}{2} \cdot \Big(\dfrac{2 - 1}{1 \cdot 2} + \dfrac{3 - 2}{2 \cdot 3} + \dfrac{4 - 3}{3 \cdot 4} + ... + \dfrac{51 - 50}{50 \cdot 51}\Big)}$

$= \sqrt{50 \cdot 51 \cdot \Big[ \Big(\dfrac{1}{1} - \dfrac{1}{2}\Big) + \Big(\dfrac{1}{2} - \dfrac{1}{3}\Big) + \Big(\dfrac{1}{3} - \dfrac{1}{4}\Big) + ... + \Big(\dfrac{1}{50} - \dfrac{1}{51}\Big)\Big]}$

$= \sqrt{50 \cdot 51 \cdot \Big(\dfrac{1}{1} - \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + ... + \dfrac{1}{50} - \dfrac{1}{51}\Big)}$

$= \sqrt{50 \cdot 51 \cdot \Big(\dfrac{1}{1} - \dfrac{1}{51}\Big)} = \sqrt{50 \cdot 51 \cdot \dfrac{51 - 1}{51}} = \sqrt{50 \cdot 50}$

$= \sqrt{ {50}^{2} } = \bf 50$