Geometric progression- a non-zero numerical sequence formed as a result of multiplying each subsequent term by a given coefficient that is not equal to zero.
Sequence definition
Before dealing with progression, one should understand the definition of a numerical sequence and the law by which it is set. Recall the natural series - the first numerical sequence that we study back in kindergarten. These are integers used to recount items. The start looks like this:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10...n
If each number of the natural series is associated with another number formed according to certain formula, we get a new sequence:
a1, a2, a3, a4, a5, a6, a7, a8, a9, a10 ... an
The number an is a common member of the sequence and the law that forms the elements of the series. Obviously, the formula for setting the natural series is simply n. For a sequence of even numbers, each element and common term is given by the formula 2n, and for odd numbers - 2n − 1.
Arithmetic and geometric progressions
Another example of exponential progression is the epidemic spread of influenza. For example, one patient per day can infect 12 people, each of the 12 will also infect another 12 people, so on the second day there will be 144 patients, on the third - 1,728, and on the fourth - 20,736.
Our program generates a geometric progression of the selected value. To do this, you will need to enter the value of the first term in the "First number" cell, the denominator of the progression in the "Difference (step)" cell, and the number of elements of the sequence in the "Last number" cell. After that, the program will provide numbers that correspond to the law of geometric progression.
Let's look at an example
Money game by mail
In Soviet times, there was a scam based on the principle of geometric progression. The essence of the scam is as follows. People received letters with 5 addresses and instructions:
- send to addresses for 1 ruble;
- cross out the first address and write your fifth;
- send invitation letters with the specified addresses to your friends and acquaintances.
The adventurers provided a logical explanation for the enrichment mechanism. Indeed, if the people invited by you send 1 ruble each, then you will return the money spent. Five invited participants in the game will send letters to their friends, in which your address is indicated at number 4. The number of such letters is already 25, and the next wave of invitees will send you a total of 25 rubles. After that, 25 people will send 5 letters each, where your address is the third and this is already 125 envelopes of 1 ruble each.
How much money did the swindlers promise at the end of the round of invitations? The answer lies in a simple geometric progression. According to their version, there will be 5 waves of invitations with your address. Since we do not take into account the unit, but start with 5 letters, the last number will be equal to 6. The first, of course, is 1. The step of our geometric progression is 5. We drive this data into the cells of the calculator and get the sequence:
1, 5, 25, 125, 625, 3125,
the sum of the sequence elements in this case is 3906. It was the profit of 3906 rubles that the scammers promised to gullible citizens. Naturally, in practice, all the money went to the organizers of the game, since at the first step the swindlers sent not one letter, but hundreds, in which their own addresses were indicated. Even if at the first step the scammers send only 200 letters, then at the fifth step 625,000 people should join the game, and the organizers will receive more than 700,000 rubles from them. Further steps no longer make sense.
Conclusion
Geometric progression is often found in reality. Use our catalog of calculators to solve interesting problems or to check case studies.
First level
Geometric progression. Comprehensive guide with examples (2019)
Numeric sequence
So let's sit down and start writing some numbers. For instance:
You can write any numbers, and there can be as many as you like (in our case, them). No matter how many numbers we write, we can always say which of them is the first, which is the second, and so on to the last, that is, we can number them. This is an example of a number sequence:
Numeric sequence is a set of numbers, each of which can be assigned a unique number.
For example, for our sequence:
The assigned number is specific to only one sequence number. In other words, there are no three second numbers in the sequence. The second number (like the -th number) is always the same.
The number with the number is called the -th member of the sequence.
We usually call the whole sequence some letter (for example,), and each member of this sequence - the same letter with an index equal to the number of this member: .
In our case:
The most common types of progression are arithmetic and geometric. In this topic, we will talk about the second kind - geometric progression.
Why do we need a geometric progression and its history.
Even in ancient times, the Italian mathematician, the monk Leonardo of Pisa (better known as Fibonacci), dealt with the practical needs of trade. The monk was faced with the task of determining what is the smallest number of weights that can be used to weigh the goods? In his writings, Fibonacci proves that such a system of weights is optimal: This is one of the first situations in which people had to deal with a geometric progression, which you have probably heard about and have at least general concept. Once you fully understand the topic, think about why such a system is optimal?
At present, in life practice, a geometric progression manifests itself when investing money in a bank, when the amount of interest is charged on the amount accumulated in the account for the previous period. In other words, if you put money on a term deposit in a savings bank, then in a year the deposit will increase by from the original amount, i.e. the new amount will be equal to the contribution multiplied by. In another year, this amount will increase by, i.е. the amount obtained at that time is again multiplied by and so on. A similar situation is described in the problems of computing the so-called compound interest- the percentage is taken each time from the amount that is on the account, taking into account the previous interest. We will talk about these tasks a little later.
There are many more simple cases where a geometric progression is applied. For example, the spread of influenza: one person infected a person, they, in turn, infected another person, and thus the second wave of infection - a person, and they, in turn, infected another ... and so on ...
By the way, a financial pyramid, the same MMM, is a simple and dry calculation according to the properties of a geometric progression. Interesting? Let's figure it out.
Geometric progression.
Let's say we have a number sequence:
You will immediately answer that it is easy and the name of such a sequence is an arithmetic progression with the difference of its members. How about something like this:
If you subtract the previous number from the next number, you will see that every time you get new difference(and so on), but the sequence definitely exists and is easy to see - each next number times more than the previous one!
This type of sequence is called geometric progression and is marked.
A geometric progression ( ) is a numerical sequence, the first term of which is different from zero, and each term, starting from the second, is equal to the previous one, multiplied by the same number. This number is called the denominator of a geometric progression.
The constraints that the first term ( ) is not equal and are not random. Let's say that there are none, and the first term is still equal, and q is, hmm .. let, then it turns out:
Agree that this is no progression.
As you understand, we will get the same results if it is any number other than zero, but. In these cases, there will simply be no progression, since the entire number series will be either all zeros, or one number, and all the rest zeros.
Now let's talk in more detail about the denominator of a geometric progression, that is, about.
Let's repeat: - this is a number, how many times does each subsequent term change geometric progression.
What do you think it could be? That's right, positive and negative, but not zero (we talked about this a little higher).
Let's say we have a positive. Let in our case, a. What is the second term and? You can easily answer that:
All right. Accordingly, if, then all subsequent members of the progression have the same sign - they positive.
What if it's negative? For example, a. What is the second term and?
It's a completely different story
Try to count the term of this progression. How much did you get? I have. Thus, if, then the signs of the terms of the geometric progression alternate. That is, if you see a progression with alternating signs in its members, then its denominator is negative. This knowledge can help you test yourself when solving problems on this topic.
Now let's practice a little: try to determine which numerical sequences are a geometric progression, and which are an arithmetic one:
Got it? Compare our answers:
- Geometric progression - 3, 6.
- Arithmetic progression - 2, 4.
- It is neither an arithmetic nor a geometric progression - 1, 5, 7.
Let's return to our last progression, and let's try to find its term in the same way as in arithmetic. As you may have guessed, there are two ways to find it.
We successively multiply each term by.
So, the -th member of the described geometric progression is equal to.
As you already guess, now you yourself will derive a formula that will help you find any member of a geometric progression. Or have you already brought it out for yourself, describing how to find the th member in stages? If so, then check the correctness of your reasoning.
Let's illustrate this by the example of finding the -th member of this progression:
In other words:
Find yourself the value of a member of a given geometric progression.
Happened? Compare our answers:
Pay attention that you got exactly the same number as in the previous method, when we successively multiplied by each previous member of the geometric progression.
Let's try to "depersonalize" this formula - we bring it into a general form and get:
The derived formula is true for all values - both positive and negative. Check it out yourself by calculating the terms of a geometric progression with following conditions: , a.
Did you count? Let's compare the results:
Agree that it would be possible to find a member of the progression in the same way as a member, however, there is a possibility of miscalculating. And if we have already found the th term of a geometric progression, a, then what could be easier than using the “truncated” part of the formula.
An infinitely decreasing geometric progression.
More recently, we talked about what can be either greater or less than zero, however, there are special values for which the geometric progression is called infinitely decreasing.
Why do you think it has such a name?
To begin with, let's write down some geometric progression consisting of members.
Let's say, then:
We see that each subsequent term is less than the previous one in times, but will there be any number? You immediately answer - "no". That is why the infinitely decreasing - decreases, decreases, but never becomes zero.
To clearly understand what this looks like visually, let's try to draw a graph of our progression. So, for our case, the formula takes the following form:
On the charts, we are accustomed to build dependence on, therefore:
The essence of the expression has not changed: in the first entry, we showed the dependence of the value of a geometric progression member on its ordinal number, and in the second entry, we simply took the value of a geometric progression member for, and the ordinal number was designated not as, but as. All that's left to do is plot the graph.
Let's see what you got. Here's the chart I got:
See? The function decreases, tends to zero, but never crosses it, so it is infinitely decreasing. Let's mark our points on the graph, and at the same time what the coordinate and means:
Try to schematically depict a graph of a geometric progression if its first term is also equal. Analyze what is the difference with our previous chart?
Did you manage? Here's the chart I got:
Now that you have fully understood the basics of the geometric progression topic: you know what it is, you know how to find its term, and you also know what an infinitely decreasing geometric progression is, let's move on to its main property.
property of a geometric progression.
Remember the property of members arithmetic progression? Yes, yes, how to find the value of a certain number of a progression when there are previous and subsequent values of the members of this progression. Remembered? This:
Now we are faced with exactly the same question for the terms of a geometric progression. To derive such a formula, let's start drawing and reasoning. You'll see, it's very easy, and if you forget, you can bring it out yourself.
Let's take another simple geometric progression, in which we know and. How to find? With an arithmetic progression, this is easy and simple, but how is it here? In fact, there is nothing complicated in geometry either - you just need to paint each value given to us according to the formula.
You ask, and now what do we do with it? Yes, very simple. To begin with, let's depict these formulas in the figure, and try to do various manipulations with them in order to come to a value.
We abstract from the numbers that we are given, we will focus only on their expression through a formula. We need to find the value highlighted orange, knowing the terms adjacent to it. Let's try to produce with them various activities, as a result of which we can get.
Addition.
Let's try to add two expressions and we get:
From given expression, as you can see, we will not be able to express in any way, therefore, we will try another option - subtraction.
Subtraction.
As you can see, we cannot express from this either, therefore, we will try to multiply these expressions by each other.
Multiplication.
Now look carefully at what we have, multiplying the terms of a geometric progression given to us in comparison with what needs to be found:
Guess what I'm talking about? Right, to find we need to take Square root from the geometric progression numbers adjacent to the desired number multiplied by each other:
Here you go. You yourself deduced the property of a geometric progression. Try writing this formula in general view. Happened?
Forgot condition when? Think about why it is important, for example, try to calculate it yourself, at. What happens in this case? That's right, complete nonsense, since the formula looks like this:
Accordingly, do not forget this limitation.
Now let's calculate what is
Correct answer - ! If you did not forget the second possible meaning, then you are a great fellow and you can immediately proceed to training, and if you forgot, read what is parsed below and pay attention to why it is necessary to write down both roots in the answer.
Let's draw both of our geometric progressions - one with a value, and the other with a value, and check if both of them have the right to exist:
In order to check whether such a geometric progression exists or not, it is necessary to see if it is the same between all its given members? Calculate q for the first and second cases.
See why we have to write two answers? Because the sign of the required term depends on whether it is positive or negative! And since we do not know what it is, we need to write both answers with a plus and a minus.
Now that you have mastered the main points and deduced the formula for the property of a geometric progression, find, knowing and
Compare your answers with the correct ones:
What do you think, what if we were given not the values of the members of the geometric progression adjacent to the desired number, but equidistant from it. For example, we need to find, and given and. Can we use the formula we derived in this case? Try to confirm or refute this possibility in the same way, describing what each value consists of, as you did when deriving the formula initially, with.
What did you get?
Now look carefully again.
and correspondingly:
From this we can conclude that the formula works not only with neighboring with the desired terms of a geometric progression, but also with equidistant from what the members are looking for.
Thus, our original formula becomes:
That is, if in the first case we said that, now we say that it can be equal to any natural number, which is less. The main thing is to be the same for both given numbers.
Practice for concrete examples just be extremely careful!
- , . Find.
- , . Find.
- , . Find.
Decided? I hope you were extremely attentive and noticed a small catch.
We compare the results.
In the first two cases, we calmly apply the above formula and get the following values:
In the third case, on closer examination serial numbers the numbers given to us, we understand that they are not equidistant from the number we are looking for: it is the previous number, but removed in position, so it is not possible to apply the formula.
How to solve it? It's actually not as difficult as it seems! Let's write down with you what each number given to us and the desired number consists of.
So we have and. Let's see what we can do with them. I suggest splitting. We get:
We substitute our data into the formula:
The next step we can find - for this we need to take the cube root of the resulting number.
Now let's look again at what we have. We have, but we need to find, and it, in turn, is equal to:
We found all the necessary data for the calculation. Substitute in the formula:
Our answer: .
Try to solve another same problem yourself:
Given: ,
Find:
How much did you get? I have - .
As you can see, in fact, you need remember only one formula- . All the rest you can withdraw without any difficulty yourself at any time. To do this, simply write the simplest geometric progression on a piece of paper and write down what, according to the above formula, each of its numbers is equal to.
The sum of the terms of a geometric progression.
Now consider the formulas that allow us to quickly calculate the sum of the terms of a geometric progression in a given interval:
To derive the formula for the sum of terms of a finite geometric progression, we multiply all parts of the above equation by. We get:
Look closely: what do the last two formulas have in common? That's right, common members, for example and so on, except for the first and last member. Let's try to subtract the 1st equation from the 2nd equation. What did you get?
Now express through the formula of a member of a geometric progression and substitute the resulting expression in our last formula:
Group the expression. You should get:
All that's left to do is express:
Accordingly, in this case.
What if? What formula works then? Imagine a geometric progression at. What is she like? Correct row same numbers, so the formula will look like this:
As with arithmetic and geometric progression, there are many legends. One of them is the legend of Seth, the creator of chess.
Many people know that the game of chess was invented in India. When the Hindu king met her, he was delighted with her wit and the variety of positions possible in her. Upon learning that it was invented by one of his subjects, the king decided to personally reward him. He called the inventor to him and ordered to ask him for whatever he wanted, promising to fulfill even the most skillful desire.
Seta asked for time to think, and when the next day Seta appeared before the king, he surprised the king with the unparalleled modesty of his request. He asked for a grain of wheat for the first square of the chessboard, wheat for the second, for the third, for the fourth, and so on.
The king was angry and drove Seth away, saying that the servant's request was unworthy of royal generosity, but promised that the servant would receive his grains for all the cells of the board.
And now the question is: using the formula for the sum of members of a geometric progression, calculate how many grains Seth should receive?
Let's start discussing. Since, according to the condition, Seth asked for a grain of wheat for the first cell of the chessboard, for the second, for the third, for the fourth, etc., we see that the problem is about a geometric progression. What is equal in this case?
Right.
Total cells of the chessboard. Respectively, . We have all the data, it remains only to substitute into the formula and calculate.
In order to give at least an approximate "scale" given number, transform using the degree properties:
Of course, if you want, you can take a calculator and calculate what kind of number you end up with, and if not, you'll have to take my word for it: the final value of the expression will be.
That is:
quintillion quadrillion trillion billion million thousand.
Fuh) If you want to imagine the enormity of this number, then estimate what size barn would be required to accommodate the entire amount of grain.
With a barn height of m and a width of m, its length would have to extend to km, i.e. twice as far as from the Earth to the Sun.
If the king were strong in mathematics, he could offer the scientist himself to count the grains, because in order to count a million grains, he would need at least a day of tireless counting, and given that it is necessary to count the quintillions, the grains would have to be counted all his life.
And now we will solve a simple problem on the sum of terms of a geometric progression.
Vasya, a 5th grade student, fell ill with the flu, but continues to go to school. Every day, Vasya infects two people who, in turn, infect two more people, and so on. Just one person in the class. In how many days will the whole class get the flu?
So, the first member of a geometric progression is Vasya, that is, a person. th member of the geometric progression, these are the two people whom he infected on the first day of his arrival. The total sum of the members of the progression is equal to the number of students 5A. Accordingly, we are talking about a progression in which:
Let's substitute our data into the formula for the sum of the terms of a geometric progression:
The whole class will get sick within days. Don't believe in formulas and numbers? Try to portray the "infection" of the students yourself. Happened? See what it looks like for me:
Calculate for yourself how many days the students would get the flu if everyone would infect a person, and there was a person in the class.
What value did you get? It turned out that everyone started to get sick after a day.
As you can see, such a task and the drawing for it resembles a pyramid, in which each subsequent “brings” new people. However, sooner or later a moment comes when the latter cannot attract anyone. In our case, if we imagine that the class is isolated, the person from closes the chain (). Thus, if a person were involved in financial pyramid, in which money was given if you bring two other participants, then the person (or in general) would not bring anyone, respectively, would lose everything that they invested in this financial scam.
Everything that was said above refers to a decreasing or increasing geometric progression, but, as you remember, we have a special kind - an infinitely decreasing geometric progression. How to calculate the sum of its members? And why does this type of progression have certain features? Let's figure it out together.
So, for starters, let's look again at this picture of an infinitely decreasing geometric progression from our example:
And now let's look at the formula for the sum of a geometric progression, derived a little earlier:
or
What are we striving for? That's right, the graph shows that it tends to zero. That is, when, it will be almost equal, respectively, when calculating the expression, we will get almost. In this regard, we believe that when calculating the sum of an infinitely decreasing geometric progression, this bracket can be neglected, since it will be equal.
- the formula is the sum of the terms of an infinitely decreasing geometric progression.
IMPORTANT! We use the formula for the sum of terms of an infinitely decreasing geometric progression only if the condition explicitly states that we need to find the sum endless the number of members.
If a specific number n is indicated, then we use the formula for the sum of n terms, even if or.
And now let's practice.
- Find the sum of the first terms of a geometric progression with and.
- Find the sum of the terms of an infinitely decreasing geometric progression with and.
I hope you were very careful. Compare our answers:
Now you know everything about geometric progression, and it's time to move from theory to practice. The most common exponential problems found on the exam are compound interest problems. It is about them that we will talk.
Problems for calculating compound interest.
You must have heard of the so-called compound interest formula. Do you understand what she means? If not, let's figure it out, because having realized the process itself, you will immediately understand what the geometric progression has to do with it.
We all go to the bank and know that there are different conditions on deposits: this is both a term, and additional maintenance, and a percentage with two different ways its calculation - simple and complex.
WITH simple interest everything is more or less clear: interest is charged once at the end of the deposit term. That is, if we are talking about putting 100 rubles a year under, then they will be credited only at the end of the year. Accordingly, by the end of the deposit, we will receive rubles.
Compound interest is an option in which interest capitalization, i.e. their addition to the amount of the deposit and the subsequent calculation of income not from the initial, but from the accumulated amount of the deposit. Capitalization does not occur constantly, but with some periodicity. As a rule, such periods are equal and most often banks use a month, a quarter or a year.
Let's say that we put all the same rubles per annum, but with a monthly capitalization of the deposit. What do we get?
Do you understand everything here? If not, let's take it step by step.
We brought rubles to the bank. By the end of the month, we should have an amount in our account consisting of our rubles plus interest on them, that is:
I agree?
We can take it out of the bracket and then we get:
Agree, this formula is already more similar to the one we wrote at the beginning. It remains to deal with percentages
In the condition of the problem, we are told about the annual. As you know, we do not multiply by - we convert percentages to decimals, that is:
Right? Now you ask, where did the number come from? Very simple!
I repeat: the condition of the problem says about ANNUAL interest accrued MONTHLY. As you know, in a year of months, respectively, the bank will charge us a part of the annual interest per month:
Realized? Now try to write what this part of the formula would look like if I said that interest is calculated daily.
Did you manage? Let's compare the results:
Well done! Let's return to our task: write how much will be credited to our account for the second month, taking into account that interest is charged on the accumulated deposit amount.
Here's what happened to me:
Or, in other words:
I think that you have already noticed a pattern and saw a geometric progression in all this. Write what its member will be equal to, or, in other words, how much money we will receive at the end of the month.
Did? Checking!
As you can see, if you put money in a bank for a year at a simple interest, then you will receive rubles, and if you put it at a compound rate, you will receive rubles. The benefit is small, but this happens only during the th year, but for a longer period, capitalization is much more profitable:
Consider another type of compound interest problem. After what you figured out, it will be elementary for you. So the task is:
Zvezda started investing in the industry in 2000 with a dollar capital. Every year since 2001, it has made a profit that is equal to the previous year's capital. How much profit will the Zvezda company receive at the end of 2003, if the profit was not withdrawn from circulation?
The capital of the Zvezda company in 2000.
- the capital of the Zvezda company in 2001.
- the capital of the Zvezda company in 2002.
- the capital of the Zvezda company in 2003.
Or we can write briefly:
For our case:
2000, 2001, 2002 and 2003.
Respectively:
rubles
Note that in this problem we do not have a division either by or by, since the percentage is given ANNUALLY and it is calculated ANNUALLY. That is, when reading the problem for compound interest, pay attention to what percentage is given, and in what period it is charged, and only then proceed to the calculations.
Now you know everything about geometric progression.
Workout.
- Find a term of a geometric progression if it is known that, and
- Find the sum of the first terms of a geometric progression, if it is known that, and
- MDM Capital started investing in the industry in 2003 with a dollar capital. Every year since 2004, she has made a profit that is equal to the previous year's capital. Company "MSK" cash flows” began investing in the industry in 2005 in the amount of $10,000, starting to make a profit from 2006 in the amount of. By how many dollars does the capital of one company exceed that of another at the end of 2007, if profits were not withdrawn from circulation?
Answers:
- Since the condition of the problem does not say that the progression is infinite and it is required to find the sum of a specific number of its members, the calculation is carried out according to the formula:
Company "MDM Capital":2003, 2004, 2005, 2006, 2007.
- increases by 100%, that is, 2 times.
Respectively:
rubles
MSK Cash Flows:2005, 2006, 2007.
- increases by, that is, times.
Respectively:
rubles
rubles
Let's summarize.
1) A geometric progression ( ) is a numerical sequence, the first term of which is different from zero, and each term, starting from the second, is equal to the previous one, multiplied by the same number. This number is called the denominator of a geometric progression.
2) The equation of the members of a geometric progression -.
3) can take any value, except for and.
- if, then all subsequent members of the progression have the same sign - they positive;
- if, then all subsequent members of the progression alternate signs;
- when - the progression is called infinitely decreasing.
4) , at - property of a geometric progression (neighboring terms)
or
, at (equidistant terms)
When you find it, do not forget that there should be two answers..
For instance,
5) The sum of the members of a geometric progression is calculated by the formula:
or
If the progression is infinitely decreasing, then:
or
IMPORTANT! We use the formula for the sum of terms of an infinitely decreasing geometric progression only if the condition explicitly states that it is necessary to find the sum of an infinite number of terms.
6) Tasks for compound interest are also calculated according to the formula of the th member of a geometric progression, provided that the funds were not withdrawn from circulation:
GEOMETRIC PROGRESSION. BRIEFLY ABOUT THE MAIN
Geometric progression( ) is a numerical sequence, the first term of which is different from zero, and each term, starting from the second, is equal to the previous one, multiplied by the same number. This number is called the denominator of a geometric progression.
Denominator of a geometric progression can take any value except for and.
- If, then all subsequent members of the progression have the same sign - they are positive;
- if, then all subsequent members of the progression alternate signs;
- when - the progression is called infinitely decreasing.
Equation of members of a geometric progression - .
The sum of the terms of a geometric progression calculated by the formula:
or
Geometric progression no less important in mathematics than in arithmetic. A geometric progression is such a sequence of numbers b1, b2,..., b[n] each next member of which is obtained by multiplying the previous one by a constant number. This number, which also characterizes the rate of growth or decrease of the progression, is called denominator of a geometric progression and denote
For complete task geometric progression, in addition to the denominator, it is necessary to know or determine its first term. For positive value denominator progression is a monotone sequence, and if this sequence of numbers is monotonically decreasing and monotonically increasing at. The case when the denominator equal to one is not considered in practice, since we have a sequence of identical numbers, and their summation is not of practical interest
General term of a geometric progression calculated according to the formula
The sum of the first n terms of a geometric progression determined by the formula
Let us consider solutions of classical geometric progression problems. Let's start with the simplest to understand.
Example 1. The first term of a geometric progression is 27, and its denominator is 1/3. Find the first six terms of a geometric progression.
Solution: We write the condition of the problem in the form
For calculations, we use the formula for the nth member of a geometric progression
Based on it, we find unknown members of the progression
As you can see, calculating the terms of a geometric progression is not difficult. The progression itself will look like this
Example 2. The first three members of a geometric progression are given: 6; -12; 24. Find the denominator and the seventh term.
Solution: We calculate the denominator of the geometric progression based on its definition
We got an alternating geometric progression whose denominator is -2. The seventh term is calculated by the formula
On this task is solved.
Example 3. A geometric progression is given by two of its members 
. Find the tenth term of the progression.
Solution:
Let's write the given values through the formulas
According to the rules, it would be necessary to find the denominator, and then look for the desired value, but for the tenth term we have
The same formula can be obtained on the basis of simple manipulations with the input data. We divide the sixth term of the series by another, as a result we get
If the resulting value is multiplied by the sixth term, we get the tenth
Thus, for such problems, with the help of simple transformations into fast way you can find the right solution.
Example 4. Geometric progression is given by recurrent formulas
Find the denominator of the geometric progression and the sum of the first six terms.
Solution:
We write the given data in the form of a system of equations
Express the denominator by dividing the second equation by the first
Find the first term of the progression from the first equation
Compute the following five terms to find the sum of the geometric progression
