Question

[Đỗ Anh Đức]

D = 3/4x5 + 3/5x6 + 3/6x7 + ..... 3/93x94 + 3/94x95

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Answer 1

[Kiều Vũ Linh]

\(D=\dfrac{3}{4\times5}+\dfrac{3}{5\times6}+\dfrac{3}{6\times7}+...+\dfrac{3}{93\times94}+\dfrac{3}{94\times95}\)

\(=3\times\left(\dfrac{1}{4\times5}+\dfrac{1}{5\times6}+\dfrac{1}{6\times7}+...+\dfrac{1}{93\times94}+\dfrac{1}{94\times95}\right)\)

\(=3\times\left(\dfrac{5-4}{4\times5}+\dfrac{6-5}{5\times6}+\dfrac{7-6}{6\times7}+...+\dfrac{94-93}{93\times94}+\dfrac{95-94}{94\times95}\right)\)

\(=3\times\left(\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+...+\dfrac{1}{93}-\dfrac{1}{94}+\dfrac{1}{94}-\dfrac{1}{95}\right)\)

\(=3\times\left(\dfrac{1}{4}-\dfrac{1}{95}\right)\)

\(=3\times\dfrac{91}{380}\)

\(=\dfrac{273}{380}\)

Answer 2

[Nguyễn Lê Nguyên Bảo]

D= \(\frac{3}{4.5}+\frac{3}{5.6}+\frac{3}{6.7}+\cdots+\frac{3}{93.94}+\frac{3}{94.95}\) ​

\(= 3 \times \left(\right. \frac{1}{4 \times 5} + \frac{1}{5 \times 6} + \frac{1}{6 \times 7} + . . . + \frac{1}{93 \times 94} + \frac{1}{94 \times 95} \left.\right)\)

\(= 3 \times \left(\right. \frac{5 - 4}{4 \times 5} + \frac{6 - 5}{5 \times 6} + \frac{7 - 6}{6 \times 7} + . . . + \frac{94 - 93}{93 \times 94} + \frac{95 - 94}{94 \times 95} \left.\right)\)

\(= 3 \times \left(\right. \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6} + \frac{1}{6} - \frac{1}{7} + . . . + \frac{1}{93} - \frac{1}{94} + \frac{1}{94} - \frac{1}{95} \left.\right)\)

\(= 3 \times \left(\right. \frac{1}{4} - \frac{1}{95} \left.\right)\)

\(= 3 \times \frac{91}{380}\)

\(= \frac{273}{380}\)