Question

16. Rezultatul calculului
1/9×22 + 1/18×33 + 1/27×44+...+1/891×1100.



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Answer 1

Răspuns:

Explicație pas cu pas:

$\dfrac{1}{9 \cdot 22} + \dfrac{1}{18 \cdot 33} + \dfrac{1}{27 \cdot 44} + ... + \dfrac{1}{891 \cdot 1100} =$

$= \dfrac{1}{9 \cdot 11} \cdot \bigg(\dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} + ... + \dfrac{1}{99 \cdot 100} \bigg)$

$= \dfrac{1}{99} \cdot \bigg(\dfrac{1}{1} - \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + ... + \dfrac{1}{99} - \dfrac{1}{100} \bigg)$

$= \dfrac{1}{99} \cdot \bigg(\dfrac{1}{1} - \dfrac{1}{100} \bigg) = \dfrac{1}{99} \cdot \dfrac{100 - 1}{100} = \dfrac{1}{99} \cdot \dfrac{99}{100}$

$= \bf \dfrac{1}{100}$