arithmetic sequence

An arithmetic sequence grows by adding the same fixed amount every step, like climbing a staircase where every stair has the same height. From 3 you go to 7, then 11, then 15 — each time adding 4.

That fixed amount is the common difference d. Because you add d once to get from the first term to the second, twice to reach the third, and (n−1) times to reach the n-th, the explicit formula is a_n = a_1 + (n − 1)d. The graph of an arithmetic sequence is a set of points lying on a straight line, and d plays the role of the slope.

The defining test is that consecutive differences are all equal: a_2 − a_1 = a_3 − a_2 = ... = d. If even one gap differs, the sequence is not arithmetic. A common slip is to write a_n = a_1 + n·d, off by one term; the correct multiplier is (n − 1).