An arithmetic sequence of numbers, sometimes alternatively called an arithmetic progression, is a sequence of numbers in which the difference between all pairs of consecutive numbers is constant. A very simple arithmetic sequence consists of the natural numbers: 1, 2, 3, 4, … where the difference between any number and the number before it is just one. 3, 7, 11, 15, 19, …. is another arithmetic sequence, but in this case the constant difference between elements is four.
A finite portion of an arithmetic sequence like 2, 3, 4 or 7, 11, 15 is called a finite arithmetic progression. To confuse matters, sometimes a finite arithmetic progression, like an arithmetic sequence, is also called an arithmetic progression. To be safe, when a progression is finite, I always say as much.
An arithmetic series is the sum of a finite arithmetic progression. An arithmetic series consisting of the first four natural numbers is 1 + 2 + 3 + 4. The sum, 10, is trivial to compute via simple addition, but for a longer series with larger numbers, having a formula to calculate the sum is indispensable.