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I understand the difference between these two terms. My textbook says that terms in a sequence follow some definite rule, or an algorithm, and it’s not always possible to express its general term via a mathematical expression.

Example : A sequence of consecutive prime numbers.

While each terms of a progression follows the same rule and we have a mathematical expression for any arbitrary term of a progression.

My problem is, I don’t fully understand this sentence terms of a progression follow the same rule.

Can they follow more than one rule? Let’s say we have a general term for a sequence

$T_n = \begin{cases} \frac{1}{n} & \text{if } n \in \{1,2,3,...10\}, \\ \frac{-1}{n} & \text{if } n \in \{11,12,.....,20\} \\ \frac{-1}{3n}& \text{if } n \in \{21,22, \ldots\} \end{cases} $

Now we can find any arbitrary term of this sequence because we have its general term (with more than a single mathematical expression). Is it a sequence or a progression?

My own understanding of sequence vs progression says that it should be a progression. If it is, can a progression have a general term which has more than one mathematical expression?

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    $\begingroup$ I don't think anyone but textbook writers worries about these distinctions. For what it's worth, when I use the term sequence, I don't imagine it to be following any definite rule or algorithm; it's just an arbitrary assignment of a mathematical object to each natrual number. $\endgroup$ Feb 1, 2020 at 10:42

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A sequence is a function whose domain is an interval of integers. The domain is often the set of natural numbers. Sequences are usually denoted using a subscript rather than in parentheses, that is, $T_n$ rather than $T(n)$.

In mathematics, the term progression is mostly used with a qualifying adjective: arithmetic progression, geometric progression, harmonic progression.

An arithmetic progression is a sequence of numbers such that the difference between any two consecutive terms is constant. A geometric progression is a sequence of numbers such that the quotient of any two consecutive terms is constant. A harmonic progression is a sequence formed by taking the inverses of an arithmetic progression.

Coming back to your examples, the sequences $\frac{1}{n}$ and $\frac{-1}{n} = \frac{1}{-n}$ and $\frac{-1}{3n} = \frac{1}{-3n}$ are harmonic progressions since $n$, $-n$ and $-3n$ are arithmetic progressions. Now you can only say that the sequence $T_n$ is the disjoint union of three harmonic progressions.

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  • $\begingroup$ Thanks for explaining it. The general term $T_n$ in my original post contains all three expressions. It is a single general term, that contains three expressions for different values of $n$. Sorry, it is my mistake. I did not use the right notation. I have not fully learnt mathjax yet, I am still learning. Now I understand that writing $T_n$ three times would make it look like three different sequences. I have edited my post $\endgroup$
    – 4d_
    Feb 1, 2020 at 11:06
  • $\begingroup$ If you wish, you could use $R_n$, $S_n$ and $T_n$ to clearly indicate you consider three different sequences. $\endgroup$
    – J.-E. Pin
    Feb 1, 2020 at 11:33
  • $\begingroup$ I understand what I did wrong. Sorry again for that. Actually that is a single sequence, not three different sequences. I am not sure whether we'd call it just a sequence, or a progression as well (based on the definition of a progression) $\endgroup$
    – 4d_
    Feb 1, 2020 at 11:42
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    $\begingroup$ I understand now. I just edited your question, please let me know if you agree. Then I will also update my answer... $\endgroup$
    – J.-E. Pin
    Feb 1, 2020 at 12:04
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    $\begingroup$ As I said, you rarely use progression without one of the qualifying adjectives I have listed in my answer. Since none of them apply to $T_n$, I would call it a sequence and just add as a comment it is a disjoint union of three HPs. $\endgroup$
    – J.-E. Pin
    Feb 4, 2020 at 5:12

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