# Properties of sequences

## Properties of sequences

Overview of Sequences

• A sequence is an ordered set of numbers where each number is called a term.

• A sequence can either be finite (having a specific number of terms) or infinite (with no end to the terms in the sequence).

• The nth term of a sequence can typically be determined using a formula.

Common Types of Sequences

• An arithmetic sequence, also known as a linear sequence, is a sequence where each term is a constant difference from the previous term.

• A geometric sequence is one in which each term is a constant multiple of the previous term.

• In a recurrence relation sequence, each term is defined as a function of its preceding term(s).

• A Fibonacci sequence is a specific type of sequence where each term is the sum of the two preceding terms.

Properties of Arithmetic Sequences

• The constant difference in an arithmetic sequence is often referred to as the common difference, denoted as ‘d’.

• The nth term of an arithmetic sequence can be identified using the equation: a + (n-1)d, where ‘a’ is the first term.

• The sum of the first ‘n’ terms in an arithmetic sequence (also known as the arithmetic series) can be found using the formula: n/2(2a + (n-1)d).

Properties of Geometric Sequences

• The constant ratio in a geometric sequence is known as the common ratio, often denoted as ‘r’.

• The nth term of a geometric sequence is typically identified by: ar^(n-1), where ‘a’ is the first term.

• The sum of the first ‘n’ terms in a geometric sequence (also known as the geometric series) can be found using the formula: a(r^n - 1) / (r - 1) for r ≠ 1. For r = 1, the sum is na.

Properties of Special Sequences

• In a Fibonacci sequence, the ratio of successive terms tends to the golden ratio (approximately 1.61803) as n approaches infinity.

• Certain types of sequences have specific properties such as palindromic sequences the terms of which read the same forwards or backwards.

Use of Sequences in Problem Solving

• Sequences can be used to solve a variety of mathematical problems including those involving series, number theory, probability, and combinatorial problems.

• Graphical representations of sequences can aid in the understanding of their properties and behaviors.

• Understanding the properties of convergent and divergent sequences is fundamental to calculus and real analysis.