Sequence, Progression and Series

Consider a set (here, repetitions of members are allowed) of numbers that are arranged according to a specific rule. If the rule of their arrangement is logical, the set of numbers is said to be a sequence. On the other hand, if the rule of their arrangement has a mathematical formula, the set of numbers is said to be a progression.
Consider the following examples :

P = {2, 3, 5, 7, 11, . . . } ; Q = {3, 1, 4, 1, 5 , 9, 2, 6, 5, . . } ; R = {2, 4, 6, 8, 10, . . .} & S = {2, 4, 8, 16, 32, 64, . . .}

The set of prime numbers (P) and the set of numbers of π (Q) are sequences but not a progressions. The members in these sets have a logic but not a mathematical formula to be predicted. The sets R and S are progressions as well as sequences.

Every progression is a sequence but the converse is not true.

When the terms of a sequence or a progression are added or subtracted, the mathematical structure so obtained is a series.

Arithmetic Series: 5 + 9 + 13 + 17 + . . .
Geometric Series: 1/2 + 1/4 + 1/8 + . .
Special series: 1.2 + 2.3 + 3.4 + . . .
Grandi's Series: 1 - 1 + 1 - 1 + 1 - 1 + . . .

Note by Shubham Poddar
1 month, 1 week ago

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