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?
Best Answer
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.