![]() ![]() Remember that in an arithmetic series, the common difference is constant and this pattern of adding and subtracting the same value as the terms are paired will continue. Thus, you have moved +2 - 2 = 0 away from 18. If you add the second term to the seventh term you will also get 18 because you are +2 and -2 away from the first and last terms. If you examine the graphic above, you can see that if you add the first and last terms you get 18. Why is Gauss' pairing up the terms in an arithmetic series The second formula is a more general formula implying n to be even or odd.Īlgebraically, both formulas are equivalent.Įxample 1: (Even Number of Terms) Find the sum of 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16.Įxample 2: (Odd Number of Terms) Find the sum of 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18. The first formula is Gauss' formula referencing n to be even. Sum, S n, of n terms of an arithmetic series. ![]() This relationship of examining a series forward and backward to determine the value of a series works for any arithmetic series. Thus, was born a formula for the sum of n terms of an arithmetic sequence. Let's generalize what Gauss actually did. In this situation, you will need to multiply the sum by the number of pairs and then divide by two, since you are actually working with 2 complete series.īy observing the series from BOTH directions simultaneously, Gauss was able to quickly solve the problem and establish a relationship that we still use today when working with arithmetic series. Simply list the ENTIRE series forward, then list the entire series in reverse and add the pairs. If the number of terms is odd, do not split the series in half. The sum of infinite terms of geometric sequence (when r < 1) is, Sn S n a / (1 - r). But what happens to the "wrapped" pairings if the series has 25 terms? Well, Gauss' discovery would need a bit of tweaking. The sum of n terms of geometric sequence is, Sn S n a (r n - 1) / (r - 1). Now, Gauss's discovery works nicely as long as you have an even number of terms in your series. Since he had 50 such pairs, he multiplied 101 times 50 and obtained the sum of the integers from 1 to 100 to be 5050. Gauss them added the paired values, noticing that the sums were all the same value (101). ![]()
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