Tính nhanh giá trị các biểu thức sau: g) G = 1 ⋅ 99 + 2 ⋅ 98 + 3 ⋅ 97 + . . . + 98 ⋅ 2 + 99 ⋅ 1 .

g) \(G = 1 \cdot 99 + 2 \cdot 98 + 3 \cdot 97 + ... + 98 \cdot 2 + 99 \cdot 1\)

\( = 1 \cdot 99 + 2 \cdot \left( {99 - 1} \right) + 3 \cdot \left( {99 - 2} \right) + ... + 98 \cdot \left( {99 - 97} \right) + 99 \cdot \left( {99 - 98} \right)\)

\[ = 1 \cdot 99 + 2 \cdot 99 - 2 \cdot 1 + 3 \cdot 99 - 3 \cdot 2 + ... + 98 \cdot 99 - 98 \cdot 97 + 99 \cdot 99 - 99 \cdot 98\]

\[ = \left( {1 \cdot 99 + 2 \cdot 99 + 3 \cdot 99 + ... + 98 \cdot 99 + 99 \cdot 99} \right) - \left( {2 \cdot 1 + 3 \cdot 2 + ... + 98 \cdot 97 + 99 \cdot 98} \right)\]

\[ = 99 \cdot \left( {1 + 2 + 3 + ... + 98 + 99} \right) - \frac{{98 \cdot 99 \cdot 100}}{3}\]                   (tương tự câu d)

\[ = 99 \cdot \frac{{99 \cdot \left( {99 + 1} \right)}}{2} - 98 \cdot 33 \cdot 100\]

\[ = 99 \cdot 99 \cdot 50 - 98 \cdot 33 \cdot 100 = 166\,\,650.\]