Types of Sequences and Series: Key Concepts with Practical Examples

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

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