Đáp án đúng: A
Phương pháp giải:
Đưa tổng đã cho về dạng: \(S = 1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{{10}} + ... + \frac{1}{{36}}.\)
+) Tính \(\frac{1}{2}S\) sau đó suy ra giá trị của biểu thức \(S.\)
Giải chi tiết:\(\begin{array}{l}S = 1 + \frac{1}{{1 + 2}} + \frac{1}{{1 + 2 + 3}} + \frac{1}{{1 + 2 + 3 + 4}} + ... + \frac{1}{{1 + 2 + 3 + 4 + ... + 8}}\\\,\,\,\,\, = 1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{{10}} + ... + \frac{1}{{36}}\\ \Rightarrow \frac{1}{2}.S = \frac{1}{2}\left( {1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{{10}} + ... + \frac{1}{{36}}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2} + \frac{1}{6} + \frac{1}{{12}} + \frac{1}{{20}} + ... + \frac{1}{{72}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + \frac{1}{{4.5}} + ... + \frac{1}{{8.9}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{8} - \frac{1}{9}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1 - \frac{1}{9} = \frac{8}{9}\\ \Rightarrow \frac{1}{2}S = \frac{8}{9}\\ \Rightarrow S = \frac{8}{9}:\frac{1}{2} = \frac{{16}}{9}.\end{array}\)
Chọn A.
