umg wrote:
What is the value of \(1^2 - 2^2 + 3^2 - 4^2 +\) ... \(+ 97^2 - 98^2 + 99^2 - 100^2\)
A. 5050
B. 5000
C. 0
D. -5000
E. -5050
Expand using formula \(a^2 - b^2\)
Solution:Let’s re-express the original expression as:
(1^2 - 2^2) + (3^2 - 4^2) + (5^2 - 6^2) + … + (97^2 - 98^2) + (99^2 - 100^2)
Note that each expression in parentheses is a difference of squares, so we can re-express as:
(1 - 2)(1 + 2) + (3 - 4)(3 + 4) + (5 - 6)(5 + 6) + … + (97 - 98)(97 + 98) + (99 -100)(99 + 100)
Now notice that the first factor of each term is -1:
(-1)(1 + 2) + (-1)(3 + 4) + (-1)(5 + 6) + … + (-1)(97 + 98) + (-1)(99 + 100)
Factoring out -1 from each term, we have:
(-1)(1 + 2 + 3 + 4 + 5 + 6 + … + 97 + 98 + 99 + 100)
We see that the second factor is the sum of the first 100 positive integers. We can evaluate this sum by using the formula: sum = average x number. The average of this evenly-spaced set is (1 + 100) / 2 = 101/2 = 50.5. We see that there are 100 numbers in the set, so we calculate that the sum of the first 100 positive integers is:
sum = average x number
sum = 50.5 x 100
sum = 5050
We recall that we factored out -1 prior to determining the sum, and we must use it to calculate the final answer. Thus, we have (-1) x 5050 = -5050.
Answer: E _________________
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