Răspuns:
a), b), c), d), e)
Explicație pas cu pas:
a)
[tex]\frac{1}{n} - \frac{1}{n + 1} = \frac{n + 1 - n}{n \cdot (n + 1)} = \red{ \bf \frac{1}{n \cdot (n + 1)}} \\ [/tex]
b)
[tex]\frac{1}{1\cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + ... + \frac{1}{2012 \cdot 2013} + \frac{1}{2013 \cdot 2014} = \\[/tex]
[tex]= \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{2012} - \frac{1}{2013} + \frac{1}{2013} - \frac{1}{2014} \\[/tex]
[tex]= \frac{1}{1} - \frac{1}{2014} = \frac{2014 - 1}{2014} = \red {\bf \frac{2013}{2014}} \\ [/tex]
c)
[tex]\frac{1}{n} - \frac{1}{n + m} = \frac{n + m - n}{n \cdot (n + m)} = \red{ \bf \frac{m}{n \cdot (n + m)}} \\ [/tex]
d)
[tex]\frac{3}{2\cdot 5} + \frac{3}{5 \cdot 8} + \frac{3}{8 \cdot 11} + ... + \frac{3}{2009 \cdot 2012} + \frac{3}{2012 \cdot 2015} = \\[/tex]
[tex]= \frac{1}{2} - \frac{1}{5} + \frac{1}{5} - \frac{1}{8} + \frac{1}{8} - \frac{1}{11} + ... + \frac{1}{2009} - \frac{1}{2012} + \frac{1}{2012} - \frac{1}{2015} \\[/tex]
[tex]= \frac{1}{2} - \frac{1}{2015} = \frac{2015 - 2}{2 \cdot 2015} = \bf \frac{2013}{4030} \\ [/tex]
e)
[tex]\frac{4}{1\cdot 5} + \frac{4}{5 \cdot 9} + \frac{4}{9 \cdot 13} + ... + \frac{4}{2009 \cdot 2013} + \frac{4}{2013 \cdot 2017} = \\[/tex]
[tex]= \frac{1}{1} - \frac{1}{5} + \frac{1}{5} - \frac{1}{9} + \frac{1}{9} - \frac{1}{13} + ... + \frac{1}{2009} - \frac{1}{2013} + \frac{1}{2013} - \frac{1}{2017} \\[/tex]
[tex]= \frac{1}{1} - \frac{1}{2017} = \frac{2017 - 1}{2017} = \bf \frac{2016}{2017} \\ [/tex]
