Question

va rog!!!!!!!!!!!!!!!

Answer 1

$\frac{1}{m} - \frac{1}{m + n} = \frac{m + n }{m(m + n)} - \frac{m}{m(m + n)} = \frac{m}{m(m - n)}$

=> Adevarat.

$a) \: \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + ... + \frac{1}{31 \times 32} \\ \\ = \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + ... + \frac{1}{31} - \frac{1}{32} \\ \\ = \frac{1}{1} - \frac{1}{32} = \frac{32}{32} - \frac{1}{32} = \bold{\frac{31}{32}} \\ \\ b) \: \frac{6}{1 \times 3} + \frac{6}{3 \times 5} + ... + \frac{6}{49 \times 51} \\ \\ = 6\left( \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + ... + \frac{1}{49 \times 51} \right) \\ \\ = 6\left( \frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + ... + \frac{1}{49} - \frac{1}{51}\right ) \\ \\ = 6\left( \frac{1}{1} - \frac{1}{51}\right ) \\ \\ = 6\left( \frac{51}{51} - \frac{1}{51} \right) \\ \\ = 6 \times \frac{50}{51} = 2 \times \frac{50}{17} = \bold{\frac{100}{17} }$