Deriving the Sum of the Arithmetic Sequence

We have discussed arithmetic progression or arithmetic sequence and you have learned how to find its nth term. In the previous post, you have also learned how to find the sum of the first n positive integers. Notice that first n positive integers 1, 2, 3, 4, 5, all the way up to n is also an arithmetic sequence with first term 1 and constant difference 1.

Now, the question that comes to mind is, how do we find the sum of an arithmetic sequence?

sum of first n integers

Recall the method that we used in “Finding the sum of the first n positive integers.” We added the integers twice with the order of the terms reversed as shown above. Clearly, we can use this method to find the sum of the arithmetic sequence 3, 7, 11, 15, 19, 23, 27. 

sum of arithmetic sequence

The sum of each term is always 30, and there are 7 terms, so 2S = 7 (30) which implies that

S = \displaystyle\frac{7(30)}{2} = 105.

The sequence above has 7 terms and 30 is the sum of the first and the last term. In a generalized arithmetic sequence, 7 is n or the number of terms  and 30 is the sum of the first term a_1 and the nth term a_n = a_1 + (n-1)d. Substituting the general arithmetic sequence to the calculation above, we have

S = \displaystyle\frac{n}{2}(a_1 + a_n) = \displaystyle\frac{n}{2} \left (a_1 + a_1 + (n-1)d \right )

which is also equal to

\displaystyle \frac{n}{2} (2a_1 + (n-1)d).

This result is also the same if we add the generalized arithmetic sequence. The terms of the generalized arithmetic sequence with first term a_1 and difference d are a_1, a_1 + d, a_1 + 2d, a_1 + 3d and so on with nth term a_1 + (n-1)d. If we add the terms as we have done above, we have the following sum.

sum of arithmetic sequence

Since there are n terms with sum 2a_1 + (n-1)d, the two sums add up to

2S = n \left ( 2a_1 + (n - 1) \right ).

Dividing both sides of the equation by 2 results to

S = \frac{n}{2} \left( 2a_1 + (n-1)d \right )

which is consistent with the result above.

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