Ta có: $A = {1^2} + {2^2} + {3^2} + ... + {9^2} + {10^2}$

$= 1\,.\,\left( {2 - 1} \right) + 2\,.\,\left( {3 - 1} \right) + 3\,.\,\left( {4 - 1} \right) + ... + 9\,.\,\left( {10 - 1} \right) + 10\,.\,\left( {11 - 1} \right)$

$= \left( {1\,.\,2 + 2\,.\,3 + 3\,.\,4 + ... + 9\,.\,10 + 10\,.\,11} \right) - \left( {1 + 2 + 3 + ... + 9 + 10} \right)$

Đặt $M = 1\,.\,2 + 2\,.\,3 + 3\,.\,4 + ... + 9\,.\,10 + 10\,.\,11$

Suy ra $3M = 1\,.\,2\,.\,3 + 2\,.\,3\,.\,3 + 3\,.\,4\,.\,3 + ... + 9\,.\,10\,.\,3 + 10\,.\,11\,.\,3$

$= 1\,.\,2\,.\,3 + 2\,.\,3\,.\,\left( {4 - 1} \right) + 3\,.\,4\,.\,\left( {5 - 1} \right) + ... + 9\,.\,10\,.\,\left( {11 - 8} \right) + 10\,.\,11\,.\,\left( {12 - 9} \right)$

$= 1\,.\,2\,.\,3 + 2\,.\,3\,.\,4 - 1\,.\,2\,.\,3 + 3\,.\,4\,.\,5 - 2\,.\,3\,.\,4 + ... + 9\,.\,10\,.\,11 - 8\,.\,9\,.\,10 + 10\,.\,11\,.\,12 - 9\,.\,10\,.\,11$

$= 10\,.\,11\,.\;12$$= 1320$.

Suy ra $M = 440$

Và $N = 1 + 2 + 3 + ... + 9 + 10$$= \frac{{\left( {10 + 1} \right)\,.\,\left( {\frac{{10 - 1}}{1} + 1} \right)}}{2}$$= 55$.

Do đó $A = M - N = 440 - 55 = 385$

Tương tự $B = {1^2} + {2^2} + ... + {5^2}$

$= 1\,.\,\left( {2 - 1} \right) + 2\,.\,\left( {3 - 1} \right) + ... +  + 5\,.\,\left( {6 - 1} \right)$

$= \left( {1\,.\,2 + 2\,.\,3 + ... + 5\,.\,6} \right) - \left( {1 + 2 + ... + 5} \right)$

Đặt $P = 1\,.\,2 + 2\,.\,3 + ... + 5\,.\,6$

$3P = 1\,.\,2\,.\,3 + 2\,.\,3\,.\,3 + ... + 5\,.\,6\,.\,3$

$= 1\,.\,2\,.\,3 + 2\,.\,3\,.\,\left( {4 - 1} \right) + ... + 5\,.\,6\,.\,\left( {7 - 4} \right)$

$= 1\,.\,2\,.\,3 + 2\,.\,3\,.\,4 - 1\,.\,2\,.\,3 + ... + 5\,.\,6\,.\,7 - 4\,.\,5\,.\,6$

$= 5\,.\,6\,.\;7$$= 210$.

Do đó $P = 70$

Và $Q = 1 + 2 + ... + 4 + 5$$= \frac{{\left( {5 + 1} \right)\,.\,\left( {\frac{{5 - 1}}{1} + 1} \right)}}{2}$$= 15$

Vậy $B = P - Q = 70 - 15 = 55$

+) $C = {2^2} + {4^2} + ... + {10^2}$

$= {2^2}\,.\,\left( {{1^2} + {2^2} + ... + {5^2}} \right)$

$= 4\,.\,B = 4\,.\,15 = 60$.

Ta tính được: $I = {1^2} - {2^2} + {3^2} - {4^2} + ..... + {9^2} - {10^2}$

$= \left( {{1^2} + {2^2} + {3^2} + {4^2} + ..... + {9^2} + {{10}^2}} \right) - 2\,.\,\left( {{2^2} + {4^2} + ..... + {{10}^2}} \right)$

$= A - 2\,.\,C$$= 385 - 2\,.\,60$$= 265$.