A sequence is a fundamental concept in mathematics. It is an ordered list of numbers or other mathematical objects, where each element is called a term. The order matters - the first term, second term, third term, and so on. For example, 2, 4, 6, 8, 10 is a sequence of even numbers.
There are several important types of sequences. Arithmetic sequences have a constant difference between consecutive terms, like 3, 7, 11, 15, where we add 4 each time. Geometric sequences have a constant ratio between consecutive terms, like 2, 6, 18, 54, where we multiply by 3 each time. Other types include Fibonacci sequences and constant sequences.
Sequences use special mathematical notation. We write the general term as a subscript n, like a-n. The subscript indicates the position in the sequence. We can write a sequence as a-1, a-2, a-3, and so on. Often we have a formula for the nth term, like a-n equals 2n plus 1, which gives us the sequence 3, 5, 7, 9.
Sequences can be visualized in several ways. We can plot them on a coordinate plane where the x-axis shows the position n and the y-axis shows the value a-n. For the sequence 2, 4, 6, 8, 10, we get points at coordinates (1,2), (2,4), (3,6), (4,8), and (5,10). The pattern becomes clear when we connect these points.
Sequences have many important applications. In mathematics, they are essential for calculus and limits. In finance, compound interest follows a geometric sequence. The Fibonacci sequence appears throughout nature. Computer algorithms often use sequences for sorting and searching. Sequences are truly fundamental building blocks in mathematics and science.