Đặt \(A=1.2+2.3+3.4+...+2015.2016\)
\(\Rightarrow3A=1.2.3+2.3.3+3.4.3+...+2015.2016.3\)
\(\Rightarrow3A=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+2015.2016.\left(2017-2014\right)\)
\(\Rightarrow3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+2015.2016.2017-2014.2015.2016\)
\(\Rightarrow3A=2015.2016.2017\)
\(\Rightarrow A=2015.2016.2017:3\)
\(\Rightarrow A=2015.672.2017\)
Vậy \(A=2015.672.2017\)
1 . 2 + 2 . 3 + 3 . 4 + ... + 2015 . 2016
3M = 1 . 2 . 3 + 2 . 3 . 3 + 3 . 4 . 3 + ... + 2015 . 2016 . 3
3M = 1 . 2 ( 3 - 0 ) + 2 . 3 ( 4 - 1 ) + 3 . 4 ( 5 - 2 ) + ... + 2015 . 2016 ( 2017 - 2014 )
3M = ( 1 . 2 . 3 + 2 . 3 . 4 + 3 . 4. 5 + ... + 2015 . 2016 . 2017 ) - ( 0 . 1 . 2 + 1 . 2 . 3 + 2 . 3 . 4 + ... + 2014 . 2015 . 2016 )
3M = 2015 . 2016 . 2017
M = \(\frac{2015.2016.2017}{3}\)
M = 2731179360
Gọi tổng 1.2+2.3+3.4+...+2015.2016 là M
\(\text{M = 1.2+2.3+3.4+...+2015.2016}\)
\(3M=1.2.3+2.3.3+3.4.3+...+2015.2016.3\)
\(3M=1.2\left(3-0\right)+2.3\left(4-1\right)+3.4\left(5-2\right)+...+2015.2016.\left(2017-2014\right)\)
\(3M=\left(1.2.3+2.3.4+3.4.5+...+2015.2016.2017\right)-\left(0.1.2+1.2.3+2.3.4+...+2014.2015.2016\right)\)
\(M=2015.2016.2017\)
\(M=\frac{2015.2016.2017}{3}\)
\(M=672.2015.2017\)
\(M=2731179360\)
Cách ngắn gọn nhất : ( 2015.2016.2017 - 2.1.0) : 3 = 2731179360\
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