The following is my interpretation of the questions inside the image:
\(\displaystyle \mbox{16. It is known that, for any natural number }\, n\, \mbox{ other than }\, 1,\, \mbox{ the following is true:}\)
. . . . .\(\displaystyle \dfrac{1}{(n\, -\, 1)(n\,+\, 1)}\, =\, \dfrac{1}{2}\, \cdot\, \left(\dfrac{1}{n\, -\, 1}\, -\, \dfrac{1}{n\, +\, 1}\right)\)
\(\displaystyle \mbox{Then the following expression is equal to which of the listed options?}\)
. . . . .\(\displaystyle \dfrac{1}{3\, \cdot\, 5}\, +\, \dfrac{1}{5\, \cdot\, 7}\, +\, \dfrac{1}{7\, \cdot\, 9}\, +\, ...\, +\, \dfrac{1}{17\, \cdot\, 19}\, +\, \dfrac{1}{19\, \cdot\, 21}\)
\(\displaystyle \mbox{A. }\, \dfrac{1}{14}\)
. . .\(\displaystyle \mbox{B. }\, \dfrac{1}{7}\)
. . .\(\displaystyle \mbox{C. }\, \dfrac{4}{21}\)
. . .\(\displaystyle \mbox{D. }\, \dfrac{2}{7}\)
. . .\(\displaystyle \mbox{E. }\, \dfrac{9}{2}\)
\(\displaystyle \mbox{17. It is know that, for any natural number }\, n,\, \mbox{ the following is true:}\)
. . . . .\(\displaystyle \dfrac{2}{n\, (n\, +\, 1)\, (n\, +\, 2)}\, =\, \dfrac{1}{n\, (n\, +\, 1)}\, -\, \dfrac{1}{(n\, +\, 1)\, (n\, +\, 2)}\)
\(\displaystyle \mbox{Then the following expression is equal to which of the listed options?}\)
. . . . .\(\displaystyle \dfrac{2}{1\, \cdot\, 2\, \cdot\, 3}\, +\, \dfrac{2}{2\, \cdot\, 3\, \cdot\, 4}\, +\, \dfrac{2}{3\, \cdot\, 4\, \cdot\, 5}\, +\, ...\, +\, \dfrac{2}{7\, \cdot\, 8\, \cdot\, 9}\, +\, \dfrac{2}{8\, \cdot\, 9\, \cdot\, 10}\)
\(\displaystyle \mbox{A. }\, \dfrac{11}{45}\)
. . .\(\displaystyle \mbox{B. }\, \dfrac{11}{72}\)
. . .\(\displaystyle \mbox{C. }\, \dfrac{22}{45}\)
. . .\(\displaystyle \mbox{D. }\, \dfrac{44}{45}\)
. . .\(\displaystyle \mbox{E. }\, \dfrac{35}{72}\)
(If not correct, please reply with corrections.)
What have you tried? How far have you gotten? Where are you stuck? Please be complete. Thank you!
