Explicație pas cu pas:
28.
[tex]a = {2}^{1} + {2}^{3} + {2}^{5} + ... + {2}^{81} = 2(1 + {2}^{2} + {2}^{4} + ... + {2}^{80})[/tex]
[tex]b = {4}^{0} + {4}^{1} + {4}^{2} + ... + {4}^{40} = {( {2}^{2})}^{0} + {( {2}^{2})}^{1} + {( {2}^{2})}^{2} + ... + {( {2}^{2})}^{40} = {2}^{0} + {2}^{2} + {2}^{4} + ... + {2}^{80} = 1 + {2}^{2} + {2}^{4} + ... + {2}^{80}[/tex]
[tex]\frac{a}{b} = \frac{2(1 + {2}^{2} + {2}^{4} + ... + {2}^{80})}{1 + {2}^{2} + {2}^{4} + ... + {2}^{80}} = \red{\bf 2} \\ [/tex]
29. imagine
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30.
[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 42} \\ = \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} \\ = \frac{1}{1} - \frac{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} \\ = \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} \\ = \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} = \bf \frac{2646}{1333} \\ [/tex]
