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