The terms sequence, series, and progression are related but have distinct meanings in mathematics. A sequence is an ordered list of numbers, a series is the sum of the terms in a sequence, and a progression is a type of sequence that follows a specific rule or pattern.
Defining the Terms
- Sequence: A sequence is an ordered list of numbers (or other elements) called terms. The order matters, and terms can repeat. Examples: 1, 2, 3, 4, 5... or 2, 4, 6, 8... or 1, 1, 2, 3, 5, 8...
- Series: A series is the sum of the terms in a sequence. Given the sequence 1, 2, 3, 4, the corresponding series is 1 + 2 + 3 + 4. This can be finite (sum of a finite number of terms) or infinite (sum of an infinite number of terms).
- Progression: A progression is a special type of sequence where the terms follow a specific rule or pattern. The two most common types are arithmetic progressions and geometric progressions.
Distinguishing Characteristics
The table below summarizes the key differences:
| Feature | Sequence | Series | Progression |
|---|---|---|---|
| Definition | Ordered list of elements | Sum of the terms in a sequence | Sequence with a specific pattern/rule |
| Operation | Listing | Addition | Listing according to a rule |
| Example | 2, 4, 6, 8... | 2 + 4 + 6 + 8 + ... | Arithmetic Progression: 2, 4, 6, 8... |
Examples
- Sequence: 1, 3, 5, 7, 9 (odd numbers). This is also an arithmetic progression.
- Series: 1 + 3 + 5 + 7 + 9 (sum of the first five odd numbers).
- Arithmetic Progression: 2, 5, 8, 11, 14 (each term increases by 3).
- Geometric Progression: 3, 6, 12, 24, 48 (each term is multiplied by 2).
Conclusion
In summary, a sequence is simply an ordered list, a series is the sum of the terms of a sequence, and a progression is a special type of sequence that follows a specific, predictable pattern. All progressions are sequences, but not all sequences are progressions. A series is derived from a sequence.
