What is a geometric progression?
In today’s article you will be able to learn about what a geometric progression is, what its general term is and you will be able to know its elabora with the different clearances.
Of course, I am going to provide you with some solved exercises so that you perro better understand the subject.
I hope you find it useful.
What is a progression?
To begin with, I perro begin by saying that you may think that A progression is an uninterrupted series or sequence of ordered numbers, which has a well-defined law that allows its formation.
The law must include a first term of the succession and a criterion that allows you to determine each of the following terms.
For example, the progression: 2, 4, 6, 8
The previous sequence has the number 2 as its first term and the following terms are formed by adding two.
By the way, the previous sequence is an arithmetic sequence.
To find the last term (I) of an arithmetic sequence or the desired term, you cánido use the following elabora:
In this article I am not going to explain much more about an arithmetic progression, but if you want to know more about the arithmetic progressionsSo, I leave you the following backlink:
What is a geometric progression?
A geometric progression or geometric sequence is a progression whose terms are integrated by quotient.
So that you cánido better understand the definition of geometric progression, I ask you to read one more time what a progression is.
Do you remember that they are governed by a law that contemplates a foreground and a criterion What determines the following terms?
Well, the quotient or division is going to help us find the constant quotient or ratio, which will help us determine each term of the sequence.
By the way, if you still don’t understand it well, don’t worry, because with the examples you will be able to understand it well.
What is the difference between an arithmetic progression and a geometric progression?
The difference that I consider to be the most important between an arithmetic progression and a geometric progression is with respect to the way in which the terms of the sequence are integrated.
An arithmetic progression is a progression whose terms integrate by difference, and a geometric progression is a progression whose terms integrate by quotient.
Examples of arithmetic progression
Some examples of arithmetic progression are:
- 4, 8, 12, 16, 20, 24
- 3, 6, 9, 12, 15, 18
- 5, 10, 15, 20, 25, 30
- 1, 2, 3, 4, 5, 6, 7, 8
Examples of geometric progression
Some examples of a geometric sequence are:
- 2, 4, 8, 12, 16, 32, 64
- 12, 36, 108, 324, 972, 2916
- 15, 75, 375, 1875, 9375, 46875
- 57, 114, 228, 456, 912, 1824, 3648
Elabora of a geometric progression
So that you perro know the last term of a geometric sequence you perro use the following elabora:
Where:
- I = the last term of the sequence.
- a = first term of the sequence.
- q = constant quotient of the sequence.
- n = number of terms in the sequence. For example, the sequence 2, 4, 8 has 3 termswhich are number 2, number 4 and number 8.
What are the clearances?
The answers to the elabora of a geometric progression are the following.
How to obtain the first term of a geometric sequence?
To obtain the first term, what you have to do is the following:
How to get the ratio of a geometric progression?
The ratio or constant quotient of a geometric progression perro be found with the following elabora:
Now, I know that you cánido find the ratio of a geometric sequence only by doing a division.
As an example I will use the following geometric sequence: 6, 18, 54, 162
Many times, what is done is to make a division between the terms of a sequence, although, it is important to see that the quotient (result of the division) is constant.
- 18 / 6 = 3
- 54 / 18 = 3
- 162 / = 3
As you cánido see, the quotient is constant, since it is always 3. Therefore, the ratio (q) is 3.
Now, if we use the elabora with the previous sequence, then, we obtain the following:
As you cánido see, also obtain that the ratio or constant quotient is equal to 3.
Solved problems of geometric progression
Next I am going to give you some solved examples of a geometric sequence.
I hope they are useful to you.
Exercise 1
What is the first term of the following sequence? __, 4, 8, 16, 32, 64
To find the first term of the sequence, what we have to do is the following:
Therefore, The first term of the sequence is 2.
Exercise 2
To verify that we did the previous exercise correctly, I am going to use the same sequence, but now we are going to look for the last term of the sequence. 2, 4, 8, 16, 32, 64
Of course, we already know what 64 is, but let’s see if 2 is really the first term of the sequence.
As you perro see, 64 is the last term of the sequence.
By the way, what you’re looking for is I (the first elabora).
If we wanted to find the next term of that sequence, then all we have to do is modify the n, that is, the number of terms that the sequence has.
Therefore, if we want to find the next term of the previous sequence, the n has to be 7.
Therefore, all you have to do is change the 6 of the previous example and put 7 in it.
In such a way that the next term of the sequence is 128.
In fact, you could search for any term you want with that elabora.
For example, if you want to find the term 100, then the n has to be equal to 100.
Exercise 3
For the following exercise we will use the following sequence: 43, 86, 172, 344, 688, 1376
Therefore, the data we have is:
- a = 43.
- I = 1376.
- n=6.
- what = ?
We don’t have the constant quotient, so let’s find it.
Therefore, the ratio of the sequence is equal to 2.
To check it, you perro divide each of its terms and it will always give you 2:
- 86 / 43 = 2
- 172 / 86 = 2
- 344 / 172 = 2
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