4.
a)
[tex] \frac{1414}{1515} - \frac{1212}{4545}= \frac{1414}{1515} - \frac{3\cdot404}{3\cdot 1515}= \frac{1414}{1515}- \frac{404}{1515}= \frac{1010}{1515}= \boxed{\frac{2}{3}} [/tex]
b)
[tex] \frac{123123}{246246} - \frac{323323}{969969}= \frac{123123}{2\cdot123123} - \frac{323323}{3\cdot969969}= \frac{1}{2} - \frac{1}{3}= \boxed{\frac{1}{6}} [/tex]
5.
a)
[tex] \frac{1}{n} - \frac{1}{n+1}= \frac{n+1}{n(n+1)}- \frac{n}{n(n+1)}= \frac{n+1-n}{n(n+1)}= \frac{1}{n(n+1)} [/tex]
b)
Folosim formula de la a) pentru a descompune fiecare fractie intr-o diferenta:
[tex]S= (\frac{1}{1} - \frac{1}{2}) +( \frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4})+...+( \frac{1}{2013} - \frac{1}{2014} ) [/tex]
Se observa ca se reduc toti termenii, mai putin primul si ultimul:
[tex]S= \frac{1}{1}- \frac{1}{2014}=\boxed{ \frac{2013}{2014}} [/tex]
6.
a)
[tex] \underbrace{ \frac{22}{55} + \frac{222}{555}+... + \frac{22222222}{55555555}}_{\text{7 termeni}}= \frac{2\cdot11}{5\cdot11}+ \frac{2\cdot111}{5\cdot111}+...+ \frac{2\cdot11111111}{5\cdot11111111}=\\\\
= \frac{2}{5} + \frac{2}{5} +...+ \frac{2}{5}= \frac{2}{5}\cdot7= \boxed{\frac{14}{5}} [/tex]
b)
[tex] \frac{360}{144} + \frac{360360}{144144} + \frac{360360360}{144144144}= \frac{360}{144} + \frac{1001\cdot360}{1001\cdot144} + \frac{1001001\cdot360}{100100\cdot144}=\\
= \frac{360}{144}\cdot3= \frac{72\cdot5}{72\cdot2}\cdot3= \frac{5}{2}\cdot3= \boxed{\frac{15}{2}} [/tex]
