Question

Tính: A = 2^3 + 4^3 + 6^3 +...+98^3 +100^3

Answer 1

\(A=2^3\left(1^3+2^3+3^3+...+50^3\right)\)

Đặt \(S=1^3+2^3+3^3+...+50^3\)

Ta có: \(n^3=\left(n-1\right)n\left(n+1\right)+n\)

Áo dụng ta được: 

\(1^3=0+1\)

\(2^3=1.2.3+2\)

\(3^3=2.3.4+3\)

...

\(50^3=49.50.51+50\)

 

\(S=0+1+1.2.3+2+2.3.4+3+...+49.50.51+50\)

\(=\left(1.2.3+2.3.4+...+49.50.51\right)+\left(1+2+...+50\right)\)

\(=\dfrac{49.50.51.52}{4}+\dfrac{50\left(50+1\right)}{2}=1625625\)

\(\Rightarrow A=2^3.1625625=13005000\)