It is a numerical sequence in which each term, starting with the second, is the result of multiplying the previous term by a constant what, called the PG reason.
Geometric Progression Example
The numerical sequence (5, 25, 125, 625...) is an increasing PG, where what=5. That is, each term of this PG, multiplied by its ratio (what=5), results in the following term.
Formula for finding the ratio (q) of a PG
Within the Crescent PG (2, 6, 18, 54...) there is a reason (what) constant yet unknown. To discover it, one must consider the terms of PG, where: (2=a1, 6=a2, 18=a3, 54=a4,...an), applying them in the following formula:
what= the2/The1
So, to find out the reason for this PG, the formula will be developed as follows: what= the2/The3 = 6/2 = 3.
The reason (what) of the PG above is 3.
Like the ratio of a PG is constant, i.e, common to all terms, we can work your formula with different terms, but always dividing it by its predecessor. Remembering that the ratio of a PG can be any rational number, excluding zero (0).
Example: what=a4/The3, which within the PG above is also found as a result what=3.
Formula to find the General Term of PG
There is a basic formula for finding any term in a PG. In the case of PG (2, 6, 18, 54, theno...), for example, where theno which can be named as the fifth or nth term, or the5, is still unknown. To find this or another term, the general formula is used:
Theno=am (what)n-m
Practical example - PG general term formula developed
it is known that:
Theno is any unknown term to be found;
Themis the first term in PG (or any other, if the first term doesn't exist);
what is the reason for PG;
Therefore, in PG (2, 6, 18, 54, theno...) where the fifth term is searched (a5), the formula will be developed as follows:
Theno=am (what)n-m
The5=a1 (q)5-1
The5=2 (3)4
The5=2.81
The5= 162
Thus, it turns out that the fifth term (the5) of PG (2, 6, 18, 54, tono...) é = 162.
It is worth remembering that it is important to find a PG's reason for finding an unknown term. In the case of PG above, for example, the ratio was already known as 3.
The Geometric Progression Rankings
Ascending Geometric Progression
For a PG to be considered increasing, its ratio will always be positive and its increasing terms, that is, they increase within the numerical sequence.
Example: (1, 4, 16, 64...), where what=4
In growing PG with positive terms, what > 1 and with negative terms 0 < what < 1.
Descending Geometric Progression
For a PG to be considered decreasing, its ratio will always be positive and different from zero and its terms decrease within the numerical sequence, that is, they decrease.
Examples: (200, 100, 50...), where what= 1/2
In descending PG with positive terms, 0 < what < 1 and with negative terms, what > 1.
Oscillating Geometric Progression
For a PG to be considered oscillating, its ratio will always be negative (what < 0) and its terms alternate between negative and positive.
Example: (-3, 6, -12, 24,...), where what = -2
Constant Geometric Progression
For a PG to be considered constant or stationary, its ratio will always be equal to one (what=1).
Example: (2, 2, 2, 2, 2...), where what=1.
Difference between Arithmetic Progression and Geometric Progression
Like PG, PA is also constituted through a numerical sequence. However, the terms of a PA are the result of the sum of each term with the reason (r), while the terms of a PG, as exemplified above, are the result of the multiplication of each term by its ratio (what).
Example:
In PA (5, 7, 9, 11, 13, 15, 17...) the reason (r) é 2. That is, the first term added to r2 results in the following term, and so on.
In PG (3, 6, 12, 24, 48, ...) the reason (what) is also 2. But in this case the term is multiplied to what 2, resulting in the following term, and so on.
See also the meaning of Arithmetic Progression.
Practical meaning of a PG: where can it be applied?
Geometric Progression allows the analysis of the decline or growth of something. In practical terms, PG enables the analysis, for example, of thermal variations, population growth, among other types of verifications present in our daily lives.