arithmetic sequence definition with examples

An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence and then subtract it from the previous one, and the result is always the same or constant then it is an arithmetic sequence.

The constant difference in all pairs of consecutive or successive numbers in a sequence is called the common difference, denoted by the letter dd. We use the common difference to go from one term to another. How? Take the current term and add the common difference to get to the next term, and so on. That is how the terms in the sequence are generated.

If the common difference between consecutive terms is positive, we say that the sequence is increasing.
On the other hand, when the difference is negative we say that the sequence is decreasing.

Sum of Arithmetic Sequence

Let us take an arithmetic sequence that has its first term to be a1 and the common difference to be d. Then the sum of the first 'n' terms of the sequence is given by

Sn = a1 + (a1 + d) + (a1 + 2d) + … + an ... (1)

Let us write the same sum from right to left (i.e., from the nth term to the first term).

Sn = an + (and) + (an – 2d) + … + a1 ... (2)

Adding (1) and (2), all terms with 'd' get canceled.

2Sn = (a1 + an) + (a1 + an) + (a1 + an) + … + (a1 + an)

2Sn = n (a1 + an)

Sn = [n(a1 + an)]/2

By substituting an = a1 + (n – 1)d into the last formula, we have

Sn = n/2 a1 + a1 + (n – 1)d

Sn = n/2 [2a1 + (n – 1)d]

Thus, we have derived both formulas for the sum of the arithmetic sequence.

Important Notes on Arithmetic Sequence:

In arithmetic sequences, the difference between every two successive numbers is the same.
The common difference of an arithmetic sequence a1, a2, a3, ... is, d = a2 - a1 = a3 - a2 = ...
The nth term of an arithmetic sequence is an = a1 + (n−1)d.
The sum of the first n terms of an arithmetic sequence is Sn = n/2[2a1 + (n − 1)d].
The common difference between arithmetic sequences can be either positive or negative or zero.

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