Xét : n³ = n³ - n² + n² - n + n = n(n² - n + n - 1) + n = n[n(n - 1) + (n - 1)] + n = (n - 1)n(n + 1) + n

Đặt S = 1³ + 2³ + 3³ + ... + n³ = (0.1.2 + 1) + (1.2.3 + 2) + (2.3.4 + 3) + ... + [(n - 1)n(n + 1) + n]

= [1.2.3 + 2.3.4 + ... + (n - 1)n(n + 1)] + (1 + 2 + 3 + ... + n)

Đặt A = 1.2.3 + 2.3.4 + ... + (n - 1)n(n + 1)

=> 4A = 1.2.3.4 + 2.3.4.(5 - 1) + ... + (n - 1)n(n + 1)[(n + 2) - (n - 2)]

          = [1.2.3.4 + 2.3.4.5 + ... + (n - 1)n(n + 1)(n + 2)] - [1.2.3.4 + ... + (n - 2)(n - 1)n(n + 1)] = (n - 1)n(n + 1)(n + 2)

=> A =\(\frac{\left(n-1\right)n\left(n+1\right)\left(n+2\right)}{4}\)

Đặt B = 1 + 2 + 3 + 4 + ... + n =\(\frac{n\left(n+1\right)}{2}\)

=> S = A + B =\(\frac{\left(n-1\right)n\left(n+1\right)\left(n+2\right)}{4}+\frac{n\left(n+1\right)}{2}=\frac{\left(n-1\right)n\left(n+1\right)\left(n+2\right)+2n\left(n+1\right)}{4}\)

\(=\frac{n\left(n+1\right)\left[2+\left(n-1\right)\left(n+2\right)\right]}{4}=\frac{n\left(n+1\right)\left(2+n^2+2n-n-2\right)}{4}\)

\(=\frac{n\left(n+1\right)\left(n^2+n\right)}{4}=\frac{n\left(n+1\right)n\left(n+1\right)}{4}=\left[\frac{n\left(n+1\right)}{2}\right]^2\)