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.