Question Video: Expressing a Given Series in Sigma Notation Mathematics • 10th Grade

Express the series 496 โˆ’ 497 + 498 โˆ’ 499 + ... โˆ’ 531 in sigma notation.

02:53

Video Transcript

Express the series 496 minus 497 plus 498 minus 499 and so on, all the way up to minus 531, in sigma notation.

We notice here that the terms in our series alternate from positive to negative. A series can be written using sigma notation as shown: the sum of the general term ๐‘ข sub ๐‘Ÿ where ๐‘Ÿ takes values from one to ๐‘›. In this question, we need to find an expression for this general term ๐‘ข sub ๐‘Ÿ as well as the value of ๐‘›. Letโ€™s begin by considering the sequence of positive integers 496, 497, 498, and so on, all the way up to 531. We can rewrite each of these terms as the constant 495 plus some integer. 496 is 495 plus one, 497 is 495 plus two, and so on, all the way up to 531 which is equal to 495 plus 36. This suggests that we have 36 terms in our sequence.

If we were looking to write this series in sigma notation, we would have the sum of 495 plus ๐‘Ÿ where ๐‘Ÿ takes values from one to 36. The series in this question is slightly more complicated, though, as we want the second, fourth, sixth, and every other even term to be negative. We can make any positive integer negative by multiplying by negative one. If we multiplied each of our terms by negative one to the power of ๐‘Ÿ, the first, third, fifth, and all our terms would be negative. As we want the even terms to be negative, we need to multiply by negative one to the power of ๐‘Ÿ plus one. The series 496 minus 497 plus 498 and so on can be written in sigma notation as negative one to the power of ๐‘Ÿ plus one multiplied by ๐‘Ÿ plus 495, where ๐‘Ÿ takes values from one to 36.

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