Ta có : \(M=1^2+2^2+3^2+.....+99^2+100^2\)
\(\Rightarrow M=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+...99\left(100-1\right)+100.\left(101-1\right)\)
\(\Rightarrow M=1.2-1+2.3-2+3.4-3+...+99.100-99+100.101-100\)
\(\Rightarrow M=\left(1.2+2.3+3.4+...+99.100+100.101\right)-\left(1+2+3+...+99+100\right)\)
Đặt \(A=1.2+2.3+3.4+....+99.100+100.101\)
\(\Rightarrow3A=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+99.100.\left(101-98\right)+100.101.\left(102-99\right)\)
\(\Rightarrow3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100+100.101.102-99.100.101\)
\(\Rightarrow3A=100.101.102\)
\(\Rightarrow A=\dfrac{100.101.102}{3}=343400\) (1)
Đặt \(B=1+2+3+....+99+100\)
\(\Rightarrow B=\left(100+1\right).100:2=5050\) (2)
Từ (1) và (2), suy ra:
\(M=A-B=343400-5050=338350\)
Vậy \(M=338350\)
~ Học tốt ~
