Ái chà , bài này dài đấy :)
Đặt A = \(1^2+2^2+3^2+...+2017^2\)
Ta thấy: 12 = 0.1 +1
22 = 1.2 +2
32 = 2.3 +3
....
20172 = 2016.2017 + 2017
\(\Rightarrow A=1+1.2+2+2.3+3+...+2016.2017+2017\)
\(A=\left(1+2+3+...+2017\right)+\left(1.2+2.3+...+2016.2017\right)\)
Đặt B = 1+2+3+...+2017
B = \(\left(2017+1\right).\left[\dfrac{2017-1}{1}+1\right]:2=2035153\)
Đặt C = 1.2+2.3+...+2016.2017
3C = 1.2.3 + 2.3.(4-1) + ... + 2016.2017. (2018-2015)
3C = 1.2.3 + 2.3.4 - 1.2.3 + ... + 2016.2017.2018 - 2015.2016.2017
3C = 2016.2017.2018 = 8205736896
C = 2735245632
Dễ thấy A = B + C = 2035153 + 2735245632
A = 2737280785
