Sequences and Series
An introduction to discrete patterns and progressions.
A sequence is an ordered list of numbers that follow a specific pattern. Each number in a sequence is called a term. We typically denote the terms of a sequence using subscript notation: Rendering....
A series is the sum of the terms of a sequence. If Rendering... is a sequence, then the corresponding series can be written with summation notation.
An arithmetic sequence is defined by a constant difference between consecutive terms. This is the common difference.
To find any term, we start at Rendering... and add the difference Rendering... for every step taken.
Notice that the number of differences added is always one less than the position of the term.
A series is the sum of the terms in a sequence. For an arithmetic progression, the sum of the first Rendering... terms is given by two equivalent forms:
We can write out the sum in two different ways