Explicație pas cu pas:
4.c.
[tex]y = \frac{1}{1\cdot 2} + \frac{1}{2 \cdot 3} + ... + \frac{1}{n \cdot (n + 1)} \\ = \frac{1}{1} - \not \frac{1}{2} + \not \frac{1}{2} - \not \frac{1}{3} + ... + \not \frac{1}{n} - \frac{1}{n + 1} \\ = \frac{1}{1} - \frac{1}{n + 1} = \frac{n + 1 - 1}{n + 1} = \bf \frac{n}{n + 1} [/tex]
5.
[tex]\frac{1}{n} - \frac{1}{n + 3} = \frac{n + 3 - n}{n \cdot (n + 3)} = \red{ \bf \frac{3}{n \cdot (n + 3)}} \\ [/tex]
[tex]S = \frac{1}{1\cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 11} + ... + \frac{1}{94 \cdot 97} + \frac{1}{97 \cdot 100} = \\ = \frac{1}{3} \cdot \Big( \frac{3}{1\cdot 4} + \frac{3}{4 \cdot 7} + \frac{3}{7 \cdot 11} + ... + \frac{3}{94 \cdot 97} + \frac{3}{97 \cdot 100}\Big)\\ = \frac{1}{3} \cdot \Big(\frac{1}{1} - \frac{1}{4} + \frac{1}{4} - \frac{1}{7} + \frac{1}{7} - \frac{1}{11} + ... + \frac{1}{94} - \frac{1}{97} + \frac{1}{97} - \frac{1}{100}\Big) \\ = \frac{1}{3} \cdot \Big(\frac{1}{1} - \frac{1}{100}\Big) = \frac{1}{3} \cdot \frac{99}{100} = \frac{33}{100} = 0,33 < 0,(3) = 0,333...[/tex]
