The main difference between a sequence and a progression lies in the existence of a specific formula: a progression adheres to a formula to calculate its nth term, while a sequence may follow a logical rule but doesn't necessarily have a formula.
Let's break this down further:
Sequences
- A sequence is an ordered list of numbers, objects, or events. The order matters.
- Sequences can be finite (ending) or infinite (never ending).
- The terms in a sequence can be related by a rule, but this rule doesn't necessarily need to be a formula.
Examples of Sequences:
- The sequence of prime numbers: 2, 3, 5, 7, 11, 13,... (This is a sequence because it's an ordered list, but there's no simple formula to directly calculate the nth prime number.)
- A sequence of squares: 1, 4, 9, 16, 25,... (Although we can describe this with a formula (n2), it's still valid as a sequence even if we just see the pattern and continue it.)
- A random sequence: 3, 1, 4, 1, 5, 9,... (This could be the digits of pi. Again, ordered but no immediately obvious formula for generating it all.)
Progressions
- A progression is a special type of sequence where the terms follow a specific mathematical formula.
- The formula allows you to calculate any term in the progression directly if you know its position.
Examples of Progressions:
-
Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant. The formula for the nth term is: an = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term number.
- Example: 2, 4, 6, 8, 10,... (a1 = 2, d = 2). So, the 10th term would be 2 + (10-1)2 = 20.
-
Geometric Progression (GP): A sequence where the ratio between consecutive terms is constant. The formula for the nth term is: an = a1 * r(n-1), where a1 is the first term, r is the common ratio, and n is the term number.
- Example: 3, 6, 12, 24, 48,... (a1 = 3, r = 2). So, the 7th term would be 3 * 2(7-1) = 192.
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Harmonic Progression (HP): A sequence where the reciprocals of the terms form an arithmetic progression. To determine terms of an HP, you work with the reciprocal's AP.
- Example: 1/2, 1/4, 1/6, 1/8, ... (The reciprocals are 2, 4, 6, 8, ... which form an AP).
Summary Table:
| Feature | Sequence | Progression |
|---|---|---|
| Definition | Ordered list of numbers, objects, or events. | Sequence with a specific formula for its terms. |
| Formula | May or may not have a formula. | Always has a specific formula. |
| Examples | Prime numbers, Random numbers, Squares. | Arithmetic, Geometric, Harmonic Progressions. |
In essence, all progressions are sequences, but not all sequences are progressions. A progression is a more restrictive definition where a formula must exist.
