Tính nhanh phép tính

Hướng Dẫn Giải

Tử số = \(\frac{{2017}}{1} + \frac{{2016}}{2} + \frac{{2015}}{3} + \ldots + \frac{1}{{2017}} + 2017\)

\( = 2017 + \frac{{2016}}{2} + \frac{{2015}}{3} + \ldots + \frac{1}{{2017}} + \underbrace {1 + 1 + \ldots + 1}_{}\)

2017 số 1

\( = (1 + 2017) + (1 + \frac{{2016}}{2}) + (1 + \frac{{2015}}{3}) + \ldots + (1 + \frac{1}{{2017}})\)

\( = 2018 + \frac{{2018}}{2} + \frac{{2018}}{3} + \ldots + \frac{{2018}}{{2017}}\)

\( = 2018 \times (1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{{2017}})\)

Do đó:

\(A = \frac{{2018 \times (1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{{2017}})}}{{1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{{2017}}}} = 2018\).