# Sequences

# Sequences

## Definition

- A
**sequence**is a list of numbers where each number is defined by a rule or pattern. **Finite sequences**have a specified number of terms, while**infinite sequences**keep going indefinitely.- Each number in a sequence is referred to as a
**term**. The position of a term in a sequence is referred to as its**index**.

## Arithmetic Sequences

- An
**Arithmetic Sequence**is one where each term after the first is found by adding a fixed or constant value, known as the**common difference**. - The formula for the nth term of an arithmetic sequence is
**a + (n - 1)d**, where**a**is the first term and**d**is the common difference.

## Geometric Sequences

- A
**Geometric Sequence**is one where each term after the first is found by multiplying the previous term by a fixed or constant value, known as the**common ratio**. - The formula for the nth term of a geometric sequence is
**ar^(n-1)**, where**a**is the first term and**r**is the common ratio.

## Fibonacci Sequences

- A
**Fibonacci Sequence**is a special type of sequence where each term is the sum of the preceding two terms. - The first two terms of a Fibonacci sequence are usually defined as 1 and 1, but the sequence can start with any two integers.

## Sequence Notations

- The nth term in a sequence is often denoted as
**u_n**or**a_n**. - The sum of the first n terms in a sequence is denoted as
**S_n**.

## Sum of Sequences

- The sum of an arithmetic sequence can be found using the formula:
**S_n = n/2(2a + (n -1)d)**. - The sum of a geometric sequence can be found using the formula:
**S_n = a(1 - r^n) / (1 - r)**for r ≠ 1. - The sum of an infinite geometric sequence exists as long as the common ratio r is between -1 and 1, and is given by the formula:
**S = a / (1 - r)**.

## Applications

- Sequences find applications in various fields of mathematics including algebra, calculus, and even advanced topics such as series solutions of differential equations.
- A thorough understanding of sequences forms the foundation for comprehending series and other complex mathematical concepts.