Arithmetic Sequences and the Common Difference

An arithmetic sequence is built by repeated addition. Start with a first term, then add the same number every time to get the next one. Because that step size never changes, plotting term number against term value gives points that line up perfectly straight — each step right moves the same amount up or down, which is exactly what a constant slope means. That constant step is called the common difference, and it is the whole personality of the sequence.

The common difference shows up as the gap between any two neighboring terms: subtract a term from the one right after it and you get the same number every time. That constancy is the test for whether something is arithmetic at all. To reach term n from term 1, you take n - 1 steps of size d, so the nth term is the first term plus (n - 1) times d. For 7, 11, 15, 19, the difference is 4, so term 20 is 7 + 19(4) = 83.

This idea sits right next to geometric sequences, where you multiply by a constant ratio instead of adding a constant difference — same structure, different operation. It also feeds directly into arithmetic series, where you add up a run of these evenly spaced terms. Spotting whether gaps are constant, growing, or shrinking is the first move before choosing which formula or sum applies.