Types of Sequences

Arithmetic Sequences: A sequence in which each term after the first is obtained by adding a constant difference to the preceding term. For example, 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.
Geometric Sequences: In this sequence, each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the common ratio. For example, 3, 6, 12, 24 is a geometric sequence with a common ratio of 2.

Sequence Notations

The nth term of a sequence is commonly written as a_n or a(n), where n is the position of the term in the sequence.
The common difference in an arithmetic sequence is often denoted as d, and the common ratio in a geometric sequence is often denoted as r.

The sum of an arithmetic series can be found using the formula: S_n = n/2 (2a + (n-1)d), where S_n represents the sum of the first n terms, a is the first term, and d is the common difference.
The sum of a geometric series can be found using the formula: S_n = a (1 - r^n ) / (1 - r) for r ≠ 1, where S_n represents the sum of the first n terms, a is the first term, and r is the common ratio. If the absolute value of r is less than 1, this is a geometric series, and the sum of an infinite number of terms, S, can be found using the formula S = a / (1 - r).

A recurrence relation for a sequence is an equation that recursively defines the sequence’s terms as functions of preceding ones. In contrast to explicit rules, in recursions the formula for the n-th term depends on one or more of the terms that came before it.
For a series with linear first order recurrence relation, the function is in the form a_n = f * a_(n-1) + c, where f and c are constants. The sequence will be a geometric sequence if c = 0 or an arithmetic sequence if f = 1. If f ≠ 1 and c ≠ 0, the sequence is a mix of an arithmetic and a geometric sequence.

A sequence is said to converge if it approaches a specific value, known as the limit, as the number of terms goes to infinity.
A sequence that does not converge is said to diverge. The sequence may go to infinity, to negative infinity, or oscillate between values.
The Limit of a Sequence can be evaluated by setting n to infinity in the expression of the sequence and simplifying.

The Fibonacci Sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. That is, after 0 and 1, each number is the sum of the two prior numbers, i.e., 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth.
This sequence is an example of a sequence defined by a recurrence relation.