Ái chà, câu này hơi dài.
\(J=\sqrt{1^3+2^3+3^3+...+100^3}\)
\(J^2=1^3+2^3+3^3+...+100^3\)
Chú ý nhé :)
Ta có: \(1^3=0+1\)
\(2^3=1.2.3+2\)
\(3^3=2.3.4+3\)
...
\(100^3=99.100.101+100\)
\(\Rightarrow J^2=\left(1+2+3+...+100\right)+\left(1.2.3+2.3.4+...+99.100.101\right)\)
Đặt \(A=\left(1+2+3+...+100\right)\)
\(A=100.\left(100+1\right):2=5050\)
Đặt \(B=1.2.3+2.3.4+...+99.100.101\)
\(\Rightarrow4B=1.2.3.4+2.3.4.\left(5-1\right)+...+99.100.101.\left(102-98\right)\)\(4B=1.2.3.4+2.3.4.5-1.2.3.4+...+99.100.101.102-98.99.100.101\)\(4B=99.100.101.102\)
\(4B=101989800\)
\(B=25497450\)
\(J^2=A+B=5050+25497450=25502500\)
\(J=\sqrt{25502500}=5050\)
Chúc bạn học tốt :))