# Types of Sequences and Series: Key Concepts with Practical Examples

Sequences and series come in various types. **Arithmetic sequences** have a constant difference between terms, while **geometric sequences** have a constant ratio. **Harmonic sequences** involve the reciprocals of integers, while **Fibonacci sequences** add the previous two terms. Series, the sum of sequence terms, follow similar classifications: **arithmetic series**, **geometric series**, and **harmonic series**. Convergence is essential for infinite series, where **convergent series** reach a limit and **divergent series** do not, shaping applications in calculus and real-world modeling.

Here’s a summary of each type of sequence with examples:

**Arithmetic Sequence**: Constant difference between terms.*Example*: \(2, 5, 8, 11, \ldots\) (difference of \(3\).**Geometric Sequence**: Constant ratio between terms.*Example*: \(3, 6, 12, 24, \ldots\) ratio of \(2\).**Harmonic Sequence**: Terms are reciprocals of positive integers.*Example*: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\)**Fibonacci Sequence**: Each term is the sum of the two previous terms.*Example*: \(0, 1, 1, 2, 3, 5, \ldots\)**Quadratic Sequence**: Second differences between terms are constant.*Example*: \(3, 7, 13, 21, \ldots\). First differences: \(4, 6, 8, \ldots\); Second differences: \(2, 2, \ldots\)**Arithmetic Series**: Sum of an arithmetic sequence.*Example*: \(2 + 5 + 8 + 11 + \ldots\)**Geometric Series**: Sum of a geometric sequence.*Example*: \(3 + 6 + 12 + 24 + \ldots\)**Harmonic Series**: Sum of a harmonic sequence.*Example*: \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots\)

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