a/b=?
a=
[tex] \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + ... + \frac{1}{42 \times 43} [/tex]
=>
[tex] \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{42} - \frac{1}{43} [/tex]
se simplifică -1/2 cu 1/2,-1/3 cu 1/3...-1/42 cu 1/42
Si va ramane
[tex] \frac{1}{1} - \frac{1}{43} [/tex]
amplificam prima fractie cu 43
[tex] \frac{43 - 1}{43} [/tex]
[tex] = \frac{42}{43} [/tex]
b.
b=
[tex] \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + ... + \frac{1}{125 \times 126} [/tex]
=>
[tex] \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + .... + \frac{1}{125} - \frac{1}{126} [/tex]
se simplifică -1/3 cu 1/3,-1/4 cu 1/4,-1/5 cu 1/5... -1/215 cu 1/215
Si va ramane
[tex] \frac{1}{2} - \frac{1}{126} [/tex]
amplificam prima fractie cu 63
=>
[tex] \frac{63 - 1}{126} [/tex]
[tex] = \frac{62}{126} [/tex]
[tex] \frac{a}{b} = ...[/tex]
[tex] \frac{a}{b} = \frac{ \frac{42}{43} }{ \frac{62}{126} } [/tex]
[tex] \frac{42}{43} \times \frac{126}{62 } = ... [/tex]
[tex] \frac{5292}{2666} [/tex]
