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Algebra Difficulty 5.1 AIME, harder Find the answer

4. (5 points) Definition: m⊗n=m×m−n×nm \otimes n=m \times m-n \times n, then 2⊗4−4⊗6−6⊗8−⋯−98⊗100=2 \otimes 4-4 \otimes 6-6 \otimes 8-\cdots-98 \otimes 100=

A number or a short expression. Spacing and $ signs are ignored.

Solution

【Solution】Solution:
 Original expression =2⊗4−4⊗6−6⊗8−⋯−98⊗100=(22−42)−(42−62)−(62−82)−⋯−(982−1002)=22−42−42+62−62+82−⋯−982+1002=1002+22−42−42=10000+4−16−16=9972 \begin{array}{l} \text { Original expression }=2 \otimes 4-4 \otimes 6-6 \otimes 8-\cdots-98 \otimes 100 \\ =\left(2^{2}-4^{2}\right)-\left(4^{2}-6^{2}\right)-\left(6^{2}-8^{2}\right)-\cdots-\left(98^{2}-100^{2}\right) \\ =2^{2}-4^{2}-4^{2}+6^{2}-6^{2}+8^{2}-\cdots-98^{2}+100^{2} \\ =100^{2}+2^{2}-4^{2}-4^{2} \\ =10000+4-16-16 \\ =9972 \end{array}

Therefore, the answer is: 9972.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.