Question

\(A=\dfrac{1.2+2.3+3.4+...+2016.2017}{2017\times2018}=....\)

Answer 1

Đối với dạng này ta dùng công thức \(a\cdot\left(a+1\right)=\dfrac{1}{3}\left[a\cdot\left(a+1\right)\cdot\left(a+2\right)-\left(a-1\right)\cdot a\cdot\left(a+1\right)\right]\)

Ta có:

\(1\cdot2=\dfrac{1}{3}\left(1\cdot2\cdot3-0\cdot1\cdot2\right)\)

\(2\cdot3=\dfrac{1}{3}\left(2\cdot3\cdot4-1\cdot2\cdot3\right)\)

$\cdots$

\(2016\cdot2017=\dfrac{1}{3}\left(2016\cdot2017\cdot2018-2015\cdot2016\cdot2017\right)\)

Cộng lại ta có: \(1\cdot 2 +2\cdot 3 +3 \cdot 4 +\cdots +2016\cdot 2017=\dfrac{1}{3} (2016\cdot 2017 \cdot 2018-0\cdot 1 \cdot 2)=\dfrac{1}{3}\cdot 2016\cdot 2017 \cdot 2018 \)

Thay vào $A$ thu được $A=672.$