Final answer:
To evaluate the given expression, we can use the difference of squares formula and the formula for sum of consecutive odd numbers. The expression evaluates to 50.
Step-by-step explanation:
To evaluate the expression 100² - 99² + 98² - 97² + 96² - 95² + ... + 2² - 1², we can notice a pattern. The terms alternate between being positive and negative, and the squared values decrease by 1 each time. We can rewrite the expression as (100² - 99²) + (98² - 97²) + (96² - 95²) + ... + (2² - 1²). Each bracketed term can be simplified using the difference of squares formula: a² - b² = (a + b)(a - b). Applying this formula to each bracketed term gives us (100 + 99)(100 - 99) + (98 + 97)(98 - 97) + (96 + 95)(96 - 95) + ... + (2 + 1)(2 - 1). Simplifying further, we get 199 + 195 + 191 + ... + 3 = 199 + 195 + ... + 3 + 1.
We can see that these are consecutive odd numbers from 199 down to 3. To find the sum of consecutive odd numbers, we can use the formula S = (n/2)(first term + last term), where n is the number of terms. In this case, we have (199 - 3)/2 + 1 = 98/2 + 1 = 49 + 1 = 50. So the final expression evaluates to 50.
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