Tính tổng F = 1² + 2² + 3² + … + n².

F = 1² + 2² + 3² + … + n²

F = 1 + (1 + 1).2 + (1 + 2).3 + (1 + 3).4 + … + (1 + n – 1)n

F = 1 + (2 + 1.2) + (3 + 2.3) + (4+ 3.4) + … + [n + (n – 1)n]

F = (1 + 2 + 3 + 4 + … + n) + [1.2 + 2.3 + 3.4 + …. + (n – 1)n]

Đặt A = 1 + 2 + 3 + 4 + … + n thì A = \(\frac{{n\left( {n + 1} \right)}}{2}\left( 1 \right)\)

Đặt B = [1.2 + 2.3 + 3.4 + …. + (n – 1)n]

Xét 3B = 1.2.3 + 2.3.3 + 3.4.3 + … + (n – 1).n.3

3B = [1.2.3 + 2.3.4 + … + (n – 1).n.(n + 1)] – (1.2.3 + 2.3.4 + … + (n – 2)(n – 1)n)

3B = (n – 1)n(n + 1)

B = \(\frac{{\left( {n--1} \right)n\left( {n + 1} \right)}}{3}\left( 2 \right)\)

Từ (1) và (2) suy ra:

F = \(\frac{{n\left( {n + 1} \right)}}{2} + \frac{{\left( {n--1} \right)n\left( {n + 1} \right)}}{3} = \frac{{3{n^2} + 3n + 2n\left( {{n^2} - 1} \right)}}{6} = \frac{{2{n^3} + 3{n^2} + n}}{6}\)

\( = \frac{{n\left( {2{n^2} + 3n + 1} \right)}}{6} = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\)

Vậy F = 1² + 2² + 3² + … + n² \( = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\).