Sequences of Numbers

A sequence is a list of numbers that follow a specific pattern. Each number in a sequence is called a term. There are different types of sequences, including arithmetic sequences, geometric sequences, and more complex sequences.

Arithmetic Sequences

In an arithmetic sequence, each term is obtained by adding a constant value to the previous term. This constant value is called the common difference. The general form of an arithmetic sequence is:

a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ...

Where a₁ is the first term and d is the common difference.

Geometric Sequences

In a geometric sequence, each term is obtained by multiplying the previous term by a constant value. This constant value is called the common ratio. The general form of a geometric sequence is:

a₁, a₁ * r, a₁ * r², a₁ * r³, ...

Where a₁ is the first term and r is the common ratio.

Finding the nth Term of a Sequence

To find the nth term of an arithmetic or geometric sequence, you can use the following formulas:

For arithmetic sequences: a_n = a₁ + (n - 1)d

For geometric sequences: a_n = a₁ * r^(n-1)

Where a_n is the nth term, a₁ is the first term, d is the common difference for arithmetic sequences, and r is the common ratio for geometric sequences.

Understanding sequences of numbers is important in mathematics and can be applied to various real-world situations, such as financial planning, population growth, and more.