Question

[Do Gia Bach]

Tính tổng

D = 1/2 + 1/14 + 1/35 + ... + 1/434.

E = 2/15 + 3/40 + 11/152 + 13/608 + 25/1824 + 30/4959.

Answer 1

[Xyz OLM]

D = \(\frac{1}{2}+\frac{1}{14}+\frac{1}{35}+...+\frac{1}{434}\)

    = \(2.\left(\frac{1}{4}+\frac{1}{28}+\frac{1}{70}+...+\frac{1}{868}\right)\)

    = \(2.\frac{1}{3}.\left(\frac{3}{4}+\frac{3}{28}+\frac{3}{70}+...+\frac{3}{868}\right)\)

    = \(\frac{2}{3}.\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{28.31}\right)\)

    = \(\frac{2}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{28}-\frac{1}{31}\right)\)

    = \(\frac{2}{3}.\left(1-\frac{1}{31}\right)\)

    = \(\frac{2}{3}.\frac{30}{31}\)

    = \(\frac{20}{31}\)

E = \(\frac{2}{15}+\frac{3}{40}+\frac{11}{152}+\frac{13}{608}+\frac{25}{1824}+\frac{30}{4959}\)

    = \(\frac{2}{3.5}+\frac{3}{5.8}+\frac{11}{8.19}+\frac{13}{19.32}+\frac{25}{32.57}+\frac{30}{57.87}\)

    = \(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+\frac{1}{11}-\frac{1}{19}+\frac{1}{19}-\frac{1}{32}+\frac{1}{32}-\frac{1}{57}+\frac{1}{57}-\frac{1}{87}\)

    = \(\frac{1}{3}-\frac{1}{87}\)

    = \(\frac{28}{87}\)