(0,5 điểm) Tính: \[I = {1^2} - {2^2} + {3^2} - {4^2} + ..... + {9^2} - {10^2}\].

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

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

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

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: \[I = {1^2} - {2^2} + {3^2} - {4^2} + ..... + {9^2} - {10^2}\]

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

\( = A - 2\,.\,C\)\( = 385 - 2\,.\,60\)\( = 265\).