S 5 = Σ n = 1 5 ( 2 n − 1 ) = + + + + = 1 + 3 + 5 + 7 + 9 = 25Īdding 5 positive odd integers, as we have done above, is managable. For example, the sum of the first 5 terms of the sequence defined by a n = 2 n − 1 follows:

is the sum of the terms of an arithmetic sequence. In some cases, the first term of an arithmetic sequence may not be given.Īn arithmetic series The sum of the terms of an arithmetic sequence. Next, use the first term a 1 = − 8 and the common difference d = 3 to find an equation for the nth term of the sequence.Ī n = − 8 + ( n − 1 ) ⋅ 3 = − 8 + 3 n − 3 = − 11 + 3 n Substitute a 1 = − 8 and a 7 = 10 into the above equation and then solve for the common difference d.

In this case, we are given the first and seventh term:Ī n = a 1 + ( n − 1 ) d U s e n = 7. In other words, find all arithmetic means between the 1 st and 7 th terms.īegin by finding the common difference d. In fact, any general term that is linear in n defines an arithmetic sequence.įind all terms in between a 1 = − 8 and a 7 = 10 of an arithmetic sequence. In general, given the first term a 1 of an arithmetic sequence and its common difference d, we can write the following:Ī 2 = a 1 + d a 3 = a 2 + d = ( a 1 + d ) + d = a 1 + 2 d a 4 = a 3 + d = ( a 1 + 2 d ) + d = a 1 + 3 d a 5 = a 4 + d = ( a 1 + 3 d ) + d = a 1 + 4 d ⋮įrom this we see that any arithmetic sequence can be written in terms of its first element, common difference, and index as follows:Ī n = a 1 + ( n − 1 ) d A r i t h m e t i c S e q u e n c e Here a 1 = 1 and the difference between any two successive terms is 2. For example, the sequence of positive odd integers is an arithmetic sequence, All sums will be 18.An arithmetic sequence A sequence of numbers where each successive number is the sum of the previous number and some constant d., or arithmetic progression Used when referring to an arithmetic sequence., is a sequence of numbers where each successive number is the sum of the previous number and some constant d.Ī n = a n − 1 + d A r i t h m e t i c S e q u e n c eĪnd because a n − a n − 1 = d, the constant d is called the common difference The constant d that is obtained from subtracting any two successive terms of an arithmetic sequence a n − a n − 1 = d. 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. But what happens to the 'wrapped' pairings if the series has 25 terms? Well, Gauss' discovery would need a bit of tweaking. 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).

Algebra 2 sequences and series formulas