Question

[Wendy ~]

Tính nhanh

\(\left(\frac{1}{4.9}+\frac{1}{9.14}+\frac{1}{14.19}+...+\frac{1}{44.49}\right).\frac{1-3-5-7-...-49}{89}\)

Answer 1

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\(S=\left(\frac{1}{4\cdot9}+\frac{1}{9\cdot14}+\frac{1}{14\cdot19}+...+\frac{1}{44\cdot49}\right)\cdot\frac{1-3-5-7-...-49}{89}\\ S=\frac{1}{5}\left(\frac{5}{4\cdot9}+\frac{5}{9\cdot14}+\frac{5}{14\cdot19}+...+\frac{5}{44\cdot49}\right)\cdot\frac{1-\left(3+5+7+...+49\right)}{89}\\ S=\frac{1}{5}\left(\frac{1}{4}-\frac{1}{9}+\frac{1}{9}-\frac{1}{14}+\frac{1}{14}-\frac{1}{19}+....+\frac{1}{44}-\frac{1}{49}\right)\cdot\frac{1-\frac{\left(49+3\right)\cdot24}{2}}{89}\\ S=\frac{1}{5}\left(\frac{1}{4}-\frac{1}{49}\right)\cdot\left(-7\right)\\ S=\frac{1}{5}\cdot\frac{45}{196}\cdot\left(-7\right)\\ S=\frac{-9}{28}\)