Question

e)$\frac{4}{1 \times 5} + \frac{4}{5 \times 9} + \frac{4}{9 \times 13} + ... + \frac{4}{2013 \times 2017}$
Urgent.Dau coroana​

Answer 1

Răspuns:

a), b), c), d), e)

Explicație pas cu pas:

a)

$\frac{1}{n} - \frac{1}{n + 1} = \frac{n + 1 - n}{n \cdot (n + 1)} = \red{ \bf \frac{1}{n \cdot (n + 1)}}$

b)

$\frac{1}{1\cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + ... + \frac{1}{2012 \cdot 2013} + \frac{1}{2013 \cdot 2014} =$

$= \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{2012} - \frac{1}{2013} + \frac{1}{2013} - \frac{1}{2014}$

$= \frac{1}{1} - \frac{1}{2014} = \frac{2014 - 1}{2014} = \red {\bf \frac{2013}{2014}}$

c)

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

d)

$\frac{3}{2\cdot 5} + \frac{3}{5 \cdot 8} + \frac{3}{8 \cdot 11} + ... + \frac{3}{2009 \cdot 2012} + \frac{3}{2012 \cdot 2015} =$

$= \frac{1}{2} - \frac{1}{5} + \frac{1}{5} - \frac{1}{8} + \frac{1}{8} - \frac{1}{11} + ... + \frac{1}{2009} - \frac{1}{2012} + \frac{1}{2012} - \frac{1}{2015}$

$= \frac{1}{2} - \frac{1}{2015} = \frac{2015 - 2}{2 \cdot 2015} = \bf \frac{2013}{4030}$

e)

$\frac{4}{1\cdot 5} + \frac{4}{5 \cdot 9} + \frac{4}{9 \cdot 13} + ... + \frac{4}{2009 \cdot 2013} + \frac{4}{2013 \cdot 2017} =$

$= \frac{1}{1} - \frac{1}{5} + \frac{1}{5} - \frac{1}{9} + \frac{1}{9} - \frac{1}{13} + ... + \frac{1}{2009} - \frac{1}{2013} + \frac{1}{2013} - \frac{1}{2017}$

$= \frac{1}{1} - \frac{1}{2017} = \frac{2017 - 1}{2017} = \bf \frac{2016}{2017}$