Sequence vs Progression 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?
 A: 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.
A: According to Marriam-Webster Dictionary

progression: a sequence of numbers in which each term is related to its predecessor by uniform law.

In this definition, Fibonacci sequences, for instance, cannot be considered progressions since each term of these sequences is dependent not only on one but on two predecessors: 
$u_{n+2}=u_{n+1}+u_{n}$

Even Quadratic sequence $1^{2}, 2^{2},3^{3} ...$ cannot be called a progression by this definition because the quadratic sequence formula, which includes only one predecessor, is not defined by any uniform law, but rather is dependent on n:
$u_{n+1}=(n+1)^{2}=n^{2}+2n+1=u_{n}+2n+1$
Therefore, the sequence $T_{n}$ in this question is no longer a progression, even though it consists of three progressions, because its terms no longer depend only on the previous term but also on n.
