Question

va rog sa le faceti pe toate​

Answer 1

Explicație pas cu pas:

observăm că putem grupa termenii, câte un termen din fiecare sumă

a)

sumele au câte: 50-3+1 = 48 termeni

$x = \dfrac{4}{3} + \dfrac{5}{4} + \dfrac{6}{5} + ... + \dfrac{51}{50} \\ y = \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5} + ... + \dfrac{1}{50}$

$x - y = \bigg(\dfrac{4}{3} - \dfrac{1}{3}\bigg) + \bigg(\dfrac{5}{4} - \dfrac{1}{4}\bigg) + \bigg(\dfrac{6}{5} - \dfrac{1}{5}\bigg) + ... + \bigg(\dfrac{51}{50} - \dfrac{1}{50}\bigg) =$

$= \dfrac{4 - 1}{3} + \dfrac{5 - 1}{4} + \dfrac{6 - 1}{5} + ... + \dfrac{51 - 1}{50}$

$= \dfrac{3}{3} + \dfrac{4}{4} + \dfrac{5}{5} + ... + \dfrac{50}{50}$

$= \underbrace{1 + 1 + 1 + ... + 1}_{48} = 48 \cdot 1 = \bf 48$

b)

sumele au câte: 30-5+1 = 26 termeni

$x = \dfrac{17}{5} + \dfrac{20}{6} + \dfrac{23}{7} + ... + \dfrac{92}{30}$

$y = \dfrac{7}{5} + \dfrac{8}{6} + \dfrac{9}{7} + ... + \dfrac{32}{30}$

$x - y = \bigg(\dfrac{17}{5} - \dfrac{7}{5}\bigg) + \bigg(\dfrac{20}{6} - \dfrac{8}{6}\bigg) + \bigg(\dfrac{23}{7} - \dfrac{9}{7}\bigg) + ... + \bigg(\dfrac{92}{30} - \dfrac{32}{30}\bigg) =$

$= \dfrac{17 - 7}{5} + \dfrac{20 - 8}{6} + \dfrac{23 - 9}{7} + ... + \dfrac{92 - 32}{30}$

$= \dfrac{10}{5} + \dfrac{12}{6} + \dfrac{14}{7} + ... + \dfrac{60}{30}$

$= \underbrace{2 + 2 + 2 + ... + 2}_{26} = 26 \cdot 2 = \bf 52$