In the text we showed how to calculate the sum of an

Chapter , Problem 13.2.45

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In the text we showed how to calculate the sum of an arithmetic series with an even number of terms. Consider the arithmetic series 5 + 12 + 19 + 26 + 33 + 40 + 47 + 54 + 61. Here, there are = 9 terms, and the difference between each term is = 7. Adding these terms directly, we find that their sum is 297. In this problem we find the sum of this arithmetic sequence in two different ways. We then use our results to obtain a general formula for the sum of an arithmetic series with an odd number of terms. (a) The sum of the first and last terms is 5+ 61 = 66, the sum of the second and next-to-last terms is 12+54 = 66, and so on. Find the sum of this arithmetic series by pairing off terms in this way. Notice that since the number of terms is odd, one of them will be unpaired. (b) This arithmetic series can be thought of as a series of eight terms (5++54) plus an additional term (61). Use the formula we found for the sum of an arithmetic series containing an even number of terms to find the sum of the given arithmetic series. (c) Find a formula for the sum of an arithmetic series with terms where is odd. Let 1 be the first term in the series, and let be the difference between consecutive terms. Show that the two approaches used in parts (a) and (b) give the same result, and show that your formula is the same as the formula given for even values of .