Question

Tính nhanh : B = 1.1+2.2+3.3+.....+100.100

Answer 1

\(B=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+....+100\left(101-1\right)\)

\(=1.2+2.3+3.4+...+100.101-\left(1+2+3+...+100\right)\)

Ta có: \(M=1.2+2.3+...+100.101\)

\(3M=1.2.\left(3-0\right)+2.3.\left(4-1\right)+...+100.101\left(102-99\right)\)

\(=-0.1.2+1.2.3-1.2.3+2.3.4-...-99.100.101+100.101.102\)

\(=100.101.102\)

\(\Rightarrow M=\frac{100.101.102}{3}=343400\)

\(N=1+2+...+100=\frac{\left(100+1\right).100}{2}=5050\)

\(B=M-N=338350\)