Explicație pas cu pas:
[tex]a = \frac{1}{1\cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + ... + \frac{1}{41 \cdot 42} + \frac{1}{42 \cdot 43} \\[/tex]
[tex]= \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{41} - \frac{1}{42} + \frac{1}{42} - \frac{1}{43} \\[/tex]
[tex]= 1 - \frac{1}{43} = \frac{43 - 1}{43} = \frac{42}{43} \\ [/tex]
[tex]b = \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + ... + \frac{1}{124 \cdot 125} + \frac{1}{125 \cdot 126} \\[/tex]
[tex]= \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} +\frac{1}{4} - \frac{1}{5} +... + \frac{1}{124} - \frac{1}{125} + \frac{1}{125} - \frac{1}{126} \\[/tex]
[tex]= \frac{1}{2} - \frac{1}{126} = \frac{63 - 1}{126} = \frac{62}{126} = \frac{31}{63} \\ [/tex]
→
[tex] \frac{a}{b} = \frac{ \frac{42}{43} }{ \frac{31}{63} } = \frac{42 \times 63}{43 \times 31} = \frac{2646}{1333} \\ [/tex]
