12 + 22 + 32 + .... + 502
= 1.(2 - 1) + 2.(3 - 1) + 3.(4 - 1) + .... + 50(51 - 1)
= 1.2 - 1 + 2.3 - 2 + 3.4 - 3 + .... + 50.51 - 50
= (1.2 + 2.3 + 3.4 + .... + 50.51) - (1 + 2 + 3 + .... + 50)
\(=\frac{50.51.52}{3}-\frac{50.51}{2}\)
= 45526
Ta có:
\(A=1^2+2^2+3^2+...+50^2\)
\(\Rightarrow A=1.1+2.2+3.3+...+50.50\)
\(\Rightarrow A=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+...+50\left(51-1\right)\)
\(\Rightarrow A=1.2-1+2.3-2+3.4-3+...+50.51-50\)
\(\Rightarrow A=\left(1.2+2.3+3.4+...+50.51\right)-\left(1+2+3+...+50\right)\)
Đặt \(B=1.2+2.3+3.4+...+50.51\)
\(\Rightarrow3B=1.2.3+2.3.3+3.4.3+...+50.51.3\)
\(\Rightarrow3B=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+50.51.\left(52-49\right)\)
\(\Rightarrow3B=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+50.51.52-49.50.51\)
\(\Rightarrow3B=50.51.52\)
\(\Rightarrow B=50.17.52\)
\(\Rightarrow B=44200\)
Đặt \(C=1+2+3+...+50\)
\(\Rightarrow C=\left(50+1\right)\left[\left(50-1\right):1+1\right]:2\)
\(\Rightarrow C=51.50:2\)
\(\Rightarrow C=1275\)
Ta có: A = B - C
\(\Rightarrow A=44200-1275\)
\(\Rightarrow A=42925\)
Vậy \(A=42925\)
