Question

x=√(1/3×5. + 1/5×7. +......1/99×101)​

Answer 1

$\text{rezultatul \: este \: x = } \frac{7 \sqrt{303} }{303}$

Answer 2

Explicație pas cu pas:

$\dfrac{1}{3 \cdot 5} = \dfrac{1}{2} \cdot \bigg(\dfrac{1}{3} - \dfrac{1}{5} \bigg)$

$x = \sqrt{\dfrac{1}{3 \cdot 5} + \dfrac{1}{5 \cdot 7} + ... + \dfrac{1}{97 \cdot 99} + \dfrac{1}{99 \cdot 101}} =$

$= \sqrt{\dfrac{1}{2} \cdot \bigg(\dfrac{1}{3} - \dfrac{1}{5} \bigg) + \dfrac{1}{2} \cdot \bigg(\dfrac{1}{5} - \dfrac{1}{7} \bigg) + ... + \dfrac{1}{2} \cdot \bigg(\dfrac{1}{97} - \dfrac{1}{99} \bigg) + \dfrac{1}{2} \cdot \bigg(\dfrac{1}{99} - \dfrac{1}{101} \bigg)}$

$= \sqrt{\dfrac{1}{2} \cdot \bigg(\dfrac{1}{3} - \dfrac{1}{5} + \dfrac{1}{5} - \dfrac{1}{7} + ... + \dfrac{1}{97} - \dfrac{1}{99} + \dfrac{1}{99} - \dfrac{1}{101} \bigg)}$

$= \sqrt{\dfrac{1}{2} \cdot \bigg(\dfrac{1}{3} - \dfrac{1}{101} \bigg)} = \sqrt{\dfrac{1}{2} \cdot \dfrac{101 - 3}{3 \cdot 101}}$

$= \sqrt{\dfrac{1}{2} \cdot \dfrac{98}{3 \cdot 101}} = \sqrt{\dfrac{49}{303}} = \bf \dfrac{7 \sqrt{303} }{303}$