Can you tell what the sequences in the image above have in common? In all of them the numbers grow according to some “logical form”. These number sequences can be classified as geometric progressions. One geometric progression (PG) is a numerical sequence in which the division of an element by the immediately preceding element always results in the same value, called a reason. Another interesting aspect that characterizes a geometric progression is that, when we choose three consecutive elements, the square of the middle element will always be equal to the product of the elements of the extremes. For example, let's look at the sequence A = (1, 2, 4, 8, 16, 32, …). We can identify the reason by choosing any element and dividing it by the immediately preceding term. Let's perform this procedure for all elements that appear in the sequence:
32 = 2, 16 = 2; 8 = 2; 4 = 2; 2 = 2
16 8 4 2 1
Therefore, the ratio of sequence A is 2. Let's see if the second rule holds. Let's choose three consecutive elements, for example,
4, 8, 16. According to the rule, the square of 8 is equal to the product of two end numbers, in this case 4 and 16. Using the potentiation properties, we have to 8² = 64. If we multiply the extremes, we get that 4 * 16 = 64. Apply these rules to other progressions and find out if the sequence is a geometric progression.Given any sequence (The1, a2, a3, a4, …, Then-1, ano, …), we can say that, be no any integer, the reason r is given by:
r = Theno
Then - 1
Let's analyze the other sequences of the initial text image, checking if they are geometric progressions.
B = {5, 25, 125, 625, 3125, …}
r = 25 = 125 = 625 = 3125 = 5
5 25 125 625
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C = {1, – 3, 9, – 27, 81, – 243, 729}
r = – 3 = 9 = – 27 = 81 = 243 = – 3
1 – 3 9 – 27 81
D=(10; 5; 2,5; 1,25; 0,625; 0,3125 …}
r = 5 = 2,5 = 1,25 = 0,625 = 0,3125 = 1
10 5 2,5 1,25 0,625 2
A geometric progression can be classified according to its reason. Let's look at the possible classifications:
If the PG presents a reason for negative value, we say it is a PG alternating or swinging, as in the example Ç. Note that a string of this type has alternating positive and negative values (1, -3, 9, -27, 81, -243, 729...);
When the first element of PG is positive and the reason r is like r > 1 or the first element of PG is negative and 0 < r < 1, we say that PG is growing. the sequences THE and B are examples of an increasing geometric progression;
If the opposite of the constant PG occurs, that is, when the first element of the PG is negative and the reason r is like r > 1 or the first element of PG is positive and 0 < r < 1, it is a PG decreasing. The sequence D is an example of a decreasing PG;
When a PG has ratio equal to 1, it is classified as a PG constant. The sequence (2, 2, 2, 2, 2, …) is a type of constant PG because its ratio is 1;
When PG has at least a null term, we say it is a geometric progression singular. We cannot determine the reason for a singular PG. An example is the sequence (2, 0, 0, 0, …).
By Amanda Gonçalves
Graduated in Mathematics
