Arithmetic and Geometric Sequences

Arithmetic and geometric sequences are two types of sequences in mathematics that follow specific patterns.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference.

The nth term of an arithmetic sequence can be calculated using the formula:

a_n = a_1 + (n - 1) * d

Where:

  • a_n is the nth term,
  • a_1 is the first term,
  • d is the common difference,
  • n is the term number.

For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3.

Geometric Sequences

A geometric sequence is a sequence of numbers in which the ratio of any two consecutive terms is constant. This ratio is called the common ratio.

The nth term of a geometric sequence can be calculated using the formula:

a_n = a_1 * r^(n - 1)

Where:

  • a_n is the nth term,
  • a_1 is the first term,
  • r is the common ratio,
  • n is the term number.

For example, in the sequence 3, 6, 12, 24, ..., the common ratio is 2.

Applications

Arithmetic and geometric sequences are used in various fields of study, including mathematics, physics, economics, and computer science. They are used to model situations where quantities increase by a fixed amount (arithmetic) or by a fixed factor (geometric). Understanding these sequences is crucial for the study of series, calculus, and many other areas of mathematics.