Sequences: Functions with fantastic properties – Some known sequences: Geometric Sequences

Hello dear friends Steemians! Continuing the series of posts on the fantastic sequences, in the present I will talk about the geometric sequences and some of its properties.

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Figure 1: Image allusive to geometric sequences

A geometric sequence or geometric progression is a sequence such that "the ratio or quotient between any two consecutive terms is constant" [1], that is, one in which any term is equal to the previous one multiplied by a constant called ratio for the progression or simply ratio.

For the geometric progression whose terms are:

it is true that: , and in general:

where: is the ratio (constant) for the progression, is the n-th term or any of the sequence, and is the pre-nth term.

The previous expression, which is the recurrence formula of a geometric progression, shows that each term of the same is obtained from the previous one, multiplying it by the reason of the sequence. It is also common to rewrite it as:

Some examples of geometric progressions are:

Example 1: The sequence of the powers of 2:

whose ratio is 2 (it is observed that when dividing two consecutive terms the result is always 2).

This sequence is of great importance in Mathematics and Computational Sciences, since it is related to the famous binary numbers (base numbering system 2).

Example 2: The sequence of the powers of 10:

whose ratio is 10 (it is observed that when dividing two consecutive terms the result is always 10).

This sequence is of importance in Mathematics and Physics, since it is related to the very useful scientific notation and, of course, to the system of decimal or base 10 numeration.

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General or n-th term of a Geometric Progression

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Sum of the first terms of a Geometric Progression

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