How to find x in geometric progression formula. Geometric progression
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Guys, today we will get acquainted with another type of progression.
The topic of today's lesson is geometric progression.
Geometric progression
Definition. A numerical sequence in which each term, starting from the second, is equal to the product of the previous one and some fixed number is called a geometric progression.Let's define our sequence recursively: $b_(1)=b$, $b_(n)=b_(n-1)*q$,
where b and q are certain given numbers. The number q is called the denominator of the progression.
Example. 1,2,4,8,16... A geometric progression in which the first term is equal to one, and $q=2$.
Example. 8,8,8,8... A geometric progression in which the first term is equal to eight,
and $q=1$.
Example. 3,-3,3,-3,3... Geometric progression in which the first term is equal to three,
and $q=-1$.
Geometric progression has the properties of monotony.
If $b_(1)>0$, $q>1$,
then the sequence is increasing.
If $b_(1)>0$, $0 The sequence is usually denoted in the form: $b_(1), b_(2), b_(3), ..., b_(n), ...$.
Just like in an arithmetic progression, if in a geometric progression the number of elements is finite, then the progression is called a finite geometric progression.
$b_(1), b_(2), b_(3), ..., b_(n-2), b_(n-1), b_(n)$.
Note that if a sequence is a geometric progression, then the sequence of squares of terms is also a geometric progression. In the second sequence, the first term is equal to $b_(1)^2$, and the denominator is equal to $q^2$.
Formula for the nth term of a geometric progression
Geometric progression can also be specified in analytical form. Let's see how to do this:$b_(1)=b_(1)$.
$b_(2)=b_(1)*q$.
$b_(3)=b_(2)*q=b_(1)*q*q=b_(1)*q^2$.
$b_(4)=b_(3)*q=b_(1)*q^3$.
$b_(5)=b_(4)*q=b_(1)*q^4$.
We easily notice the pattern: $b_(n)=b_(1)*q^(n-1)$.
Our formula is called the "formula of the nth term of a geometric progression."
Let's return to our examples.
Example. 1,2,4,8,16... Geometric progression in which the first term is equal to one,
and $q=2$.
$b_(n)=1*2^(n)=2^(n-1)$.
Example. 16,8,4,2,1,1/2… A geometric progression in which the first term is equal to sixteen, and $q=\frac(1)(2)$.
$b_(n)=16*(\frac(1)(2))^(n-1)$.
Example. 8,8,8,8... A geometric progression in which the first term is equal to eight, and $q=1$.
$b_(n)=8*1^(n-1)=8$.
Example. 3,-3,3,-3,3... A geometric progression in which the first term is equal to three, and $q=-1$.
$b_(n)=3*(-1)^(n-1)$.
Example. Given a geometric progression $b_(1), b_(2), …, b_(n), … $.
a) It is known that $b_(1)=6, q=3$. Find $b_(5)$.
b) It is known that $b_(1)=6, q=2, b_(n)=768$. Find n.
c) It is known that $q=-2, b_(6)=96$. Find $b_(1)$.
d) It is known that $b_(1)=-2, b_(12)=4096$. Find q.
Solution.
a) $b_(5)=b_(1)*q^4=6*3^4=486$.
b) $b_n=b_1*q^(n-1)=6*2^(n-1)=768$.
$2^(n-1)=\frac(768)(6)=128$, since $2^7=128 => n-1=7; n=8$.
c) $b_(6)=b_(1)*q^5=b_(1)*(-2)^5=-32*b_(1)=96 => b_(1)=-3$.
d) $b_(12)=b_(1)*q^(11)=-2*q^(11)=4096 => q^(11)=-2048 => q=-2$.
Example. The difference between the seventh and fifth terms of the geometric progression is 192, the sum of the fifth and sixth terms of the progression is 192. Find the tenth term of this progression.
Solution.
We know that: $b_(7)-b_(5)=192$ and $b_(5)+b_(6)=192$.
We also know: $b_(5)=b_(1)*q^4$; $b_(6)=b_(1)*q^5$; $b_(7)=b_(1)*q^6$.
Then:
$b_(1)*q^6-b_(1)*q^4=192$.
$b_(1)*q^4+b_(1)*q^5=192$.
We received a system of equations:
$\begin(cases)b_(1)*q^4(q^2-1)=192\\b_(1)*q^4(1+q)=192\end(cases)$.
Equating our equations we get:
$b_(1)*q^4(q^2-1)=b_(1)*q^4(1+q)$.
$q^2-1=q+1$.
$q^2-q-2=0$.
We got two solutions q: $q_(1)=2, q_(2)=-1$.
Substitute sequentially into the second equation:
$b_(1)*2^4*3=192 => b_(1)=4$.
$b_(1)*(-1)^4*0=192 =>$ no solutions.
We got that: $b_(1)=4, q=2$.
Let's find the tenth term: $b_(10)=b_(1)*q^9=4*2^9=2048$.
Sum of a finite geometric progression
Let us have a finite geometric progression. Let's, just like for an arithmetic progression, calculate the sum of its terms.Let a finite geometric progression be given: $b_(1),b_(2),…,b_(n-1),b_(n)$.
Let us introduce the designation for the sum of its terms: $S_(n)=b_(1)+b_(2)+⋯+b_(n-1)+b_(n)$.
In the case when $q=1$. All terms of the geometric progression are equal to the first term, then it is obvious that $S_(n)=n*b_(1)$.
Let us now consider the case $q≠1$.
Let's multiply the above amount by q.
$S_(n)*q=(b_(1)+b_(2)+⋯+b_(n-1)+b_(n))*q=b_(1)*q+b_(2)*q+⋯ +b_(n-1)*q+b_(n)*q=b_(2)+b_(3)+⋯+b_(n)+b_(n)*q$.
Note:
$S_(n)=b_(1)+(b_(2)+⋯+b_(n-1)+b_(n))$.
$S_(n)*q=(b_(2)+⋯+b_(n-1)+b_(n))+b_(n)*q$.
$S_(n)*q-S_(n)=(b_(2)+⋯+b_(n-1)+b_(n))+b_(n)*q-b_(1)-(b_(2 )+⋯+b_(n-1)+b_(n))=b_(n)*q-b_(1)$.
$S_(n)(q-1)=b_(n)*q-b_(1)$.
$S_(n)=\frac(b_(n)*q-b_(1))(q-1)=\frac(b_(1)*q^(n-1)*q-b_(1)) (q-1)=\frac(b_(1)(q^(n)-1))(q-1)$.
$S_(n)=\frac(b_(1)(q^(n)-1))(q-1)$.
We have obtained the formula for the sum of a finite geometric progression.
Example.
Find the sum of the first seven terms of a geometric progression whose first term is 4 and the denominator is 3.
Solution.
$S_(7)=\frac(4*(3^(7)-1))(3-1)=2*(3^(7)-1)=4372$.
Example.
Find the fifth term of the geometric progression that is known: $b_(1)=-3$; $b_(n)=-3072$; $S_(n)=-4095$.
Solution.
$b_(n)=(-3)*q^(n-1)=-3072$.
$q^(n-1)=1024$.
$q^(n)=1024q$.
$S_(n)=\frac(-3*(q^(n)-1))(q-1)=-4095$.
$-4095(q-1)=-3*(q^(n)-1)$.
$-4095(q-1)=-3*(1024q-1)$.
$1365q-1365=1024q-1$.
$341q=$1364.
$q=4$.
$b_5=b_1*q^4=-3*4^4=-3*256=-768$.
Characteristic property of geometric progression
Guys, a geometric progression is given. Let's look at its three consecutive members: $b_(n-1),b_(n),b_(n+1)$.We know that:
$\frac(b_(n))(q)=b_(n-1)$.
$b_(n)*q=b_(n+1)$.
Then:
$\frac(b_(n))(q)*b_(n)*q=b_(n)^(2)=b_(n-1)*b_(n+1)$.
$b_(n)^(2)=b_(n-1)*b_(n+1)$.
If the progression is finite, then this equality holds for all terms except the first and last.
If it is not known in advance what form the sequence has, but it is known that: $b_(n)^(2)=b_(n-1)*b_(n+1)$.
Then we can safely say that this is a geometric progression.
A number sequence is a geometric progression only when the square of each member is equal to the product of the two adjacent members of the progression. Do not forget that for a finite progression this condition is not satisfied for the first and last terms.
Let's look at this identity: $\sqrt(b_(n)^(2))=\sqrt(b_(n-1)*b_(n+1))$.
$|b_(n)|=\sqrt(b_(n-1)*b_(n+1))$.
$\sqrt(a*b)$ is called the geometric mean of the numbers a and b.
The modulus of any term of a geometric progression is equal to the geometric mean of its two neighboring terms.
Example.
Find x such that $x+2; 2x+2; 3x+3$ were three consecutive terms of a geometric progression.
Solution.
Let's use the characteristic property:
$(2x+2)^2=(x+2)(3x+3)$.
$4x^2+8x+4=3x^2+3x+6x+6$.
$x^2-x-2=0$.
$x_(1)=2$ and $x_(2)=-1$.
Let us sequentially substitute our solutions into the original expression:
With $x=2$, we got the sequence: 4;6;9 – a geometric progression with $q=1.5$.
For $x=-1$, we get the sequence: 1;0;0.
Answer: $x=2.$
Problems to solve independently
1. Find the eighth first term of the geometric progression 16;-8;4;-2….2. Find the tenth term of the geometric progression 11,22,44….
3. It is known that $b_(1)=5, q=3$. Find $b_(7)$.
4. It is known that $b_(1)=8, q=-2, b_(n)=512$. Find n.
5. Find the sum of the first 11 terms of the geometric progression 3;12;48….
6. Find x such that $3x+4; 2x+4; x+5$ are three consecutive terms of a geometric progression.
Purpose of the lesson: to introduce students to a new type of sequence - an infinitely decreasing geometric progression.
Tasks:
formulating an initial idea of the limit of a numerical sequence;
acquaintance with another way to convert infinite periodic fractions into ordinary ones using the formula for the sum of an infinitely decreasing geometric progression;
development of intellectual qualities of schoolchildren’s personality such as logical thinking, ability to make evaluative actions, and generalization;
fostering activity, mutual assistance, collectivism, and interest in the subject.
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Lesson on the topic “Infinitely decreasing geometric progression” (algebra, 10th grade)
The purpose of the lesson: introducing students to a new type of sequence - an infinitely decreasing geometric progression.
Tasks:
formulating an initial idea of the limit of a numerical sequence; acquaintance with another way to convert infinite periodic fractions into ordinary ones using the formula for the sum of an infinitely decreasing geometric progression;
development of intellectual qualities of schoolchildren’s personality such as logical thinking, ability to make evaluative actions, and generalization;
fostering activity, mutual assistance, collectivism, and interest in the subject.
Equipment: computer class, projector, screen.
Lesson type: lesson - learning a new topic.
During the classes
I. Org. moment. State the topic and purpose of the lesson.
II. Updating students' knowledge.
In 9th grade you studied arithmetic and geometric progressions.
Questions
1. Definition of arithmetic progression.
(An arithmetic progression is a sequence in which each member
Starting from the second, it is equal to the previous term added to the same number).
2. Formula n th term of an arithmetic progression
3. Formula for the sum of the first n terms of an arithmetic progression.
( or )
4. Definition of geometric progression.
(A geometric progression is a sequence of non-zero numbers
Each term of which, starting from the second, is equal to the previous term multiplied by
Same number).
5. Formula n th term of the geometric progression
6. Formula for the sum of the first n members of a geometric progression.
7. What other formulas do you know?
(, Where ; ;
; , )
Tasks
1. Arithmetic progression is given by the formula a n = 7 – 4n . Find a 10. (-33)
2. In arithmetic progression a 3 = 7 and a 5 = 1 . Find a 4 . (4)
3. In arithmetic progression a 3 = 7 and a 5 = 1 . Find a 17 . (-35)
4. In arithmetic progression a 3 = 7 and a 5 = 1 . Find S 17. (-187)
5. For geometric progressionfind the fifth term.
6. For geometric progression find the nth term.
7. Exponentially b 3 = 8 and b 5 = 2. Find b 4 . (4)
8. Exponentially b 3 = 8 and b 5 = 2. Find b 1 and q.
9. Exponentially b 3 = 8 and b 5 = 2. Find S5. (62)
III. Learning a new topic(demonstration of presentation).
Consider a square with a side equal to 1. Let's draw another square whose side is half the size of the first square, then another one whose side is half the second, then the next one, etc. Each time the side of the new square is equal to half of the previous one.
As a result, we received a sequence of sides of squaresforming a geometric progression with the denominator.
And, what is very important, the more we build such squares, the smaller the side of the square will be. For example ,
Those. As the number n increases, the terms of the progression approach zero.
Using this figure, you can consider another sequence.
For example, the sequence of areas of squares:
And, again, if n increases indefinitely, then the area approaches zero as close as you like.
Let's look at another example. An equilateral triangle with sides equal to 1 cm. Let's construct the following triangle with the vertices in the midpoints of the sides of the 1st triangle, according to the theorem about the midline of the triangle - the side of the 2nd is equal to half the side of the first, the side of the 3rd is equal to half the side of the 2nd, etc. Again we obtain a sequence of lengths of the sides of triangles.
At .
If we consider a geometric progression with a negative denominator.
Then, again, with increasing numbers n terms of the progression approach zero.
Let's pay attention to the denominators of these sequences. Everywhere the denominators were less than 1 in absolute value.
We can conclude: a geometric progression will be infinitely decreasing if the modulus of its denominator is less than 1.
Frontal work.
Definition:
A geometric progression is said to be infinitely decreasing if the modulus of its denominator is less than one..
Using the definition, you can decide whether a geometric progression is infinitely decreasing or not.
Task
Is the sequence an infinitely decreasing geometric progression if it is given by the formula:
Solution:
Let's find q.
; ; ; .
this geometric progression is infinitely decreasing.
b) this sequence is not an infinitely decreasing geometric progression.
Consider a square with a side equal to 1. Divide it in half, one of the halves in half, etc. The areas of all the resulting rectangles form an infinitely decreasing geometric progression:
The sum of the areas of all rectangles obtained in this way will be equal to the area of the 1st square and equal to 1.
But on the left side of this equality is the sum of an infinite number of terms.
Let's consider the sum of the first n terms.
According to the formula for the sum of the first n terms of a geometric progression, it is equal to.
If n increases without limit, then
or . Therefore, i.e. .
Sum of an infinitely decreasing geometric progressionthere is a sequence limit S 1, S 2, S 3, …, S n, ….
For example, for progression,
we have
Because
The sum of an infinitely decreasing geometric progressioncan be found using the formula.
III. Comprehension and consolidation(completing tasks).
№13; №14; №15(1,3); №16(1,3); №18(1,3); №19; №20.
IV. Summarizing.
What sequence did you get acquainted with today?
Define an infinitely decreasing geometric progression.
How to prove that a geometric progression is infinitely decreasing?
Give the formula for the sum of an infinitely decreasing geometric progression.
V. Homework.
2. № 15(2,4); №16(2,4); 18(2,4).
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Slide captions:
Everyone should be able to think consistently, judge with evidence, and refute incorrect conclusions: a physicist and a poet, a tractor driver and a chemist. E. Kolman In mathematics, one should remember not the formulas, but the processes of thinking. V.P. Ermakov It is easier to find the squaring of a circle than to outwit a mathematician. Augustus de Morgan What science could be more noble, more admirable, more useful to humanity than mathematics? Franklin
Infinitely decreasing geometric progression grade 10
I. Arithmetic and geometric progressions. Questions 1. Definition of arithmetic progression. An arithmetic progression is a sequence in which each term, starting from the second, is equal to the previous term added to the same number. 2. Formula for the nth term of an arithmetic progression. 3. Formula for the sum of the first n terms of an arithmetic progression. 4. Definition of geometric progression. A geometric progression is a sequence of non-zero numbers, each term of which, starting from the second, is equal to the previous term multiplied by the same number 5. Formula for the nth term of a geometric progression. 6. Formula for the sum of the first n terms of a geometric progression.
II. Arithmetic progression. Tasks An arithmetic progression is given by the formula a n = 7 – 4 n Find a 10 . (-33) 2. In arithmetic progression, a 3 = 7 and a 5 = 1. Find a 4 . (4) 3. In arithmetic progression a 3 = 7 and a 5 = 1. Find a 17 . (-35) 4. In arithmetic progression, a 3 = 7 and a 5 = 1. Find S 17. (-187)
II. Geometric progression. Tasks 5. For a geometric progression, find the fifth term 6. For a geometric progression, find the nth term. 7. In geometric progression b 3 = 8 and b 5 = 2. Find b 4 . (4) 8. In geometric progression b 3 = 8 and b 5 = 2. Find b 1 and q. 9. In geometric progression b 3 = 8 and b 5 = 2. Find S5. (62)
definition: A geometric progression is called infinitely decreasing if the modulus of its denominator is less than one.
Problem No. 1 Is the sequence an infinitely decreasing geometric progression if it is given by the formula: Solution: a) this geometric progression is infinitely decreasing. b) this sequence is not an infinitely decreasing geometric progression.
The sum of an infinitely decreasing geometric progression is the limit of the sequence S 1, S 2, S 3, ..., S n, .... For example, for the progression we have Since the Sum of an infinitely decreasing geometric progression can be found using the formula
Completing tasks Find the sum of an infinitely decreasing geometric progression with the first term 3, the second 0.3. 2. No. 13; No. 14; textbook, p. 138 3. No. 15(1;3); No.16(1;3) No.18(1;3); 4. No. 19; No. 20.
What sequence did you get acquainted with today? Define an infinitely decreasing geometric progression. How to prove that a geometric progression is infinitely decreasing? Give the formula for the sum of an infinitely decreasing geometric progression. Questions
The famous Polish mathematician Hugo Steinhaus jokingly claims that there is a law that is formulated as follows: a mathematician will do it better. Namely, if you entrust two people, one of whom is a mathematician, to perform any work unfamiliar to them, then the result will always be the following: the mathematician will do it better. Hugo Steinhaus 01/14/1887-02/25/1972
Let's consider a certain series.
7 28 112 448 1792...
It is absolutely clear that the value of any of its elements is exactly four times greater than the previous one. This means that this series is a progression.
A geometric progression is an infinite sequence of numbers, the main feature of which is that the next number is obtained from the previous one by multiplying by a specific number. This is expressed by the following formula.
a z +1 =a z ·q, where z is the number of the selected element.
Accordingly, z ∈ N.
The period when geometric progression is studied at school is 9th grade. Examples will help you understand the concept:
0.25 0.125 0.0625...
Based on this formula, the denominator of the progression can be found as follows:
Neither q nor b z can be zero. Also, each of the elements of the progression should not be equal to zero.
Accordingly, to find out the next number in a series, you need to multiply the last one by q.
To set this progression, you must specify its first element and denominator. After this, it is possible to find any of the subsequent terms and their sum.
Varieties
Depending on q and a 1, this progression is divided into several types:
- If both a 1 and q are greater than one, then such a sequence is a geometric progression increasing with each subsequent element. An example of this is presented below.
Example: a 1 =3, q=2 - both parameters are greater than one.
Then the number sequence can be written like this:
3 6 12 24 48 ...
- If |q| is less than one, that is, multiplication by it is equivalent to division, then a progression with similar conditions is a decreasing geometric progression. An example of this is presented below.
Example: a 1 =6, q=1/3 - a 1 is greater than one, q is less.
Then the number sequence can be written as follows:
6 2 2/3 ... - any element is 3 times larger than the element following it.
- Alternating sign. If q<0, то знаки у чисел последовательности постоянно чередуются вне зависимости от a 1 , а элементы ни возрастают, ни убывают.
Example: a 1 = -3, q = -2 - both parameters are less than zero.
Then the number sequence can be written like this:
3, 6, -12, 24,...
Formulas
There are many formulas for convenient use of geometric progressions:
- Z-term formula. Allows you to calculate an element under a specific number without calculating previous numbers.
Example:q = 3, a 1 = 4. It is required to count the fourth element of the progression.
Solution:a 4 = 4 · 3 4-1 = 4 · 3 3 = 4 · 27 = 108.
- The sum of the first elements whose quantity is equal to z. Allows you to calculate the sum of all elements of a sequence up toa zinclusive.
Since (1-q) is in the denominator, then (1 - q)≠ 0, therefore q is not equal to 1.
Note: if q=1, then the progression would be a series of infinitely repeating numbers.
Sum of geometric progression, examples:a 1 = 2, q= -2. Calculate S5.
Solution:S 5 = 22 - calculation using the formula.
- Amount if |q| < 1 и если z стремится к бесконечности.
Example:a 1 = 2 , q= 0.5. Find the amount.
Solution:S z = 2 · = 4
S z = 2 + 1 + 0.5 + 0.25 + 0.125 + 0.0625 = 3.9375 4
Some properties:
- Characteristic property. If the following condition works for anyz, then the given number series is a geometric progression:
a z 2 = a z -1 · az+1
- Also, the square of any number in a geometric progression is found by adding the squares of any two other numbers in a given series, if they are equidistant from this element.
a z 2 = a z - t 2 + a z + t 2 , Wheret- the distance between these numbers.
- Elementsdiffer in qonce.
- The logarithms of the elements of a progression also form a progression, but an arithmetic one, that is, each of them is greater than the previous one by a certain number.
Examples of some classic problems
To better understand what a geometric progression is, examples with solutions for class 9 can help.
- Conditions:a 1 = 3, a 3 = 48. Findq.
Solution: each subsequent element is greater than the previous one inq once.It is necessary to express some elements in terms of others using a denominator.
Hence,a 3 = q 2 · a 1
When substitutingq= 4
- Conditions:a 2 = 6, a 3 = 12. Calculate S 6.
Solution:To do this, just find q, the first element and substitute it into the formula.
a 3 = q· a 2 , hence,q= 2
a 2 = q · a 1 ,That's why a 1 = 3
S 6 = 189
- · a 1 = 10, q= -2. Find the fourth element of the progression.
Solution: to do this, it is enough to express the fourth element through the first and through the denominator.
a 4 = q 3· a 1 = -80
Application example:
- A bank client made a deposit in the amount of 10,000 rubles, under the terms of which every year the client will have 6% of it added to the principal amount. How much money will be in the account after 4 years?
Solution: The initial amount is 10 thousand rubles. This means that a year after the investment the account will have an amount equal to 10,000 + 10,000 · 0.06 = 10000 1.06
Accordingly, the amount in the account after another year will be expressed as follows:
(10000 · 1.06) · 0.06 + 10000 · 1.06 = 1.06 · 1.06 · 10000
That is, every year the amount increases by 1.06 times. This means that to find the amount of funds in the account after 4 years, it is enough to find the fourth element of the progression, which is given by the first element equal to 10 thousand and the denominator equal to 1.06.
S = 1.06 1.06 1.06 1.06 10000 = 12625
Examples of sum calculation problems:
Geometric progression is used in various problems. An example for finding the sum can be given as follows:
a 1 = 4, q= 2, calculateS 5.
Solution: all the data necessary for the calculation is known, you just need to substitute them into the formula.
S 5 = 124
- a 2 = 6, a 3 = 18. Calculate the sum of the first six elements.
Solution:
In geom. progression, each next element is q times greater than the previous one, that is, to calculate the sum you need to know the elementa 1 and denominatorq.
a 2 · q = a 3
q = 3
Similarly, you need to finda 1 , knowinga 2 Andq.
a 1 · q = a 2
a 1 =2
S 6 = 728.
NUMERIC SEQUENCES VI
§ l48. Sum of an infinitely decreasing geometric progression
Until now, when talking about sums, we have always assumed that the number of terms in these sums is finite (for example, 2, 15, 1000, etc.). But when solving some problems (especially higher mathematics) one has to deal with the sums of an infinite number of terms
S= a 1 + a 2 + ... + a n + ... . (1)
What are these amounts? A-priory the sum of an infinite number of terms a 1 , a 2 , ..., a n , ... is called the limit of the sum S n first P numbers when P -> ∞ :
S=S n = (a 1 + a 2 + ... + a n ). (2)
Limit (2), of course, may or may not exist. Accordingly, they say that the sum (1) exists or does not exist.
How can we find out whether sum (1) exists in each specific case? The general solution to this issue goes far beyond the scope of our program. However, there is one important special case that we must now consider. We will talk about summing the terms of an infinitely decreasing geometric progression.
Let a 1 , a 1 q , a 1 q 2, ... is an infinitely decreasing geometric progression. This means that | q |< 1. Сумма первых P terms of this progression is equal
From the basic theorems on the limits of variables (see § 136) we obtain:
But 1 = 1, a qn = 0. Therefore
So, the sum of an infinitely decreasing geometric progression is equal to the first term of this progression divided by one minus the denominator of this progression.
1) The sum of the geometric progression 1, 1/3, 1/9, 1/27, ... is equal to
and the sum of the geometric progression is 12; -6; 3; - 3 / 2 , ... equal
2) Convert a simple periodic fraction 0.454545 ... into an ordinary one.
To solve this problem, imagine this fraction as an infinite sum:
The right side of this equality is the sum of an infinitely decreasing geometric progression, the first term of which is equal to 45/100, and the denominator is 1/100. That's why
Using the described method, a general rule for converting simple periodic fractions into ordinary fractions can be obtained (see Chapter II, § 38):
To convert a simple periodic fraction into an ordinary fraction, you need to do the following: in the numerator put the period of the decimal fraction, and in the denominator - a number consisting of nines taken as many times as there are digits in the period of the decimal fraction.
3) Convert the mixed periodic fraction 0.58333 .... into an ordinary fraction.
Let's imagine this fraction as an infinite sum:
On the right side of this equality, all terms, starting from 3/1000, form an infinitely decreasing geometric progression, the first term of which is equal to 3/1000, and the denominator is 1/10. That's why
Using the described method, a general rule for converting mixed periodic fractions into ordinary fractions can be obtained (see Chapter II, § 38). We deliberately do not present it here. There is no need to remember this cumbersome rule. It is much more useful to know that any mixed periodic fraction can be represented as the sum of an infinitely decreasing geometric progression and a certain number. And the formula
for the sum of an infinitely decreasing geometric progression, you must, of course, remember.
As an exercise, we suggest that you, in addition to the problems No. 995-1000 given below, once again turn to problem No. 301 § 38.
Exercises
995. What is called the sum of an infinitely decreasing geometric progression?
996. Find the sums of infinitely decreasing geometric progressions:
997. At what values X progression
is it infinitely decreasing? Find the sum of such a progression.
998. In an equilateral triangle with side A a new triangle is inscribed by connecting the midpoints of its sides; a new triangle is inscribed in this triangle in the same way, and so on ad infinitum.
a) the sum of the perimeters of all these triangles;
b) the sum of their areas.
999. Square with side A a new square is inscribed by connecting the midpoints of its sides; a square is inscribed in this square in the same way, and so on ad infinitum. Find the sum of the perimeters of all these squares and the sum of their areas.
1000. Compose an infinitely decreasing geometric progression such that its sum is equal to 25/4, and the sum of the squares of its terms is equal to 625/24.
Geometric progression, along with arithmetic progression, is an important number series that is studied in the school algebra course in the 9th grade. In this article we will look at the denominator of a geometric progression and how its value affects its properties.
Definition of geometric progression
First, let's give the definition of this number series. A geometric progression is a series of rational numbers that is formed by sequentially multiplying its first element by a constant number called the denominator.
For example, the numbers in the series 3, 6, 12, 24, ... are a geometric progression, because if you multiply 3 (the first element) by 2, you get 6. If you multiply 6 by 2, you get 12, and so on.
The members of the sequence under consideration are usually denoted by the symbol ai, where i is an integer indicating the number of the element in the series.
The above definition of progression can be written in mathematical language as follows: an = bn-1 * a1, where b is the denominator. It's easy to check this formula: if n = 1, then b1-1 = 1, and we get a1 = a1. If n = 2, then an = b * a1, and we again come to the definition of the series of numbers in question. Similar reasoning can be continued for large values of n.
Denominator of geometric progression
The number b completely determines what character the entire number series will have. The denominator b can be positive, negative, or greater than or less than one. All of the above options lead to different sequences:
- b > 1. There is an increasing series of rational numbers. For example, 1, 2, 4, 8, ... If element a1 is negative, then the entire sequence will increase only in absolute value, but decrease depending on the sign of the numbers.
- b = 1. Often this case is not called a progression, since there is an ordinary series of identical rational numbers. For example, -4, -4, -4.
Formula for amount
Before moving on to the consideration of specific problems using the denominator of the type of progression under consideration, an important formula for the sum of its first n elements should be given. The formula looks like: Sn = (bn - 1) * a1 / (b - 1).
You can obtain this expression yourself if you consider the recursive sequence of terms of the progression. Also note that in the above formula it is enough to know only the first element and the denominator to find the sum of an arbitrary number of terms.
Infinitely decreasing sequence
An explanation was given above of what it is. Now, knowing the formula for Sn, let's apply it to this number series. Since any number whose modulus does not exceed 1 tends to zero when raised to large powers, that is, b∞ => 0 if -1
Since the difference (1 - b) will always be positive, regardless of the value of the denominator, the sign of the sum of an infinitely decreasing geometric progression S∞ is uniquely determined by the sign of its first element a1.
Now let's look at several problems where we will show how to apply the acquired knowledge on specific numbers.
Task No. 1. Calculation of unknown elements of progression and sum
Given a geometric progression, the denominator of the progression is 2, and its first element is 3. What will its 7th and 10th terms be equal to, and what is the sum of its seven initial elements?
The condition of the problem is quite simple and involves the direct use of the above formulas. So, to calculate element number n, we use the expression an = bn-1 * a1. For the 7th element we have: a7 = b6 * a1, substituting the known data, we get: a7 = 26 * 3 = 192. We do the same for the 10th term: a10 = 29 * 3 = 1536.
Let's use the well-known formula for the sum and determine this value for the first 7 elements of the series. We have: S7 = (27 - 1) * 3 / (2 - 1) = 381.
Problem No. 2. Determining the sum of arbitrary elements of a progression
Let -2 be equal to the denominator of the geometric progression bn-1 * 4, where n is an integer. It is necessary to determine the sum from the 5th to the 10th element of this series, inclusive.
The problem posed cannot be solved directly using known formulas. It can be solved using 2 different methods. For completeness of presentation of the topic, we present both.
Method 1. The idea is simple: you need to calculate the two corresponding sums of the first terms, and then subtract the other from one. We calculate the smaller amount: S10 = ((-2)10 - 1) * 4 / (-2 - 1) = -1364. Now we calculate the larger sum: S4 = ((-2)4 - 1) * 4 / (-2 - 1) = -20. Note that in the last expression only 4 terms were summed, since the 5th is already included in the amount that needs to be calculated according to the conditions of the problem. Finally, we take the difference: S510 = S10 - S4 = -1364 - (-20) = -1344.
Method 2. Before substituting numbers and counting, you can obtain a formula for the sum between the m and n terms of the series in question. We do exactly the same as in method 1, only we first work with the symbolic representation of the amount. We have: Snm = (bn - 1) * a1 / (b - 1) - (bm-1 - 1) * a1 / (b - 1) = a1 * (bn - bm-1) / (b - 1). You can substitute known numbers into the resulting expression and calculate the final result: S105 = 4 * ((-2)10 - (-2)4) / (-2 - 1) = -1344.
Problem No. 3. What is the denominator?
Let a1 = 2, find the denominator of the geometric progression, provided that its infinite sum is 3, and it is known that this is a decreasing series of numbers.
Based on the conditions of the problem, it is not difficult to guess which formula should be used to solve it. Of course, for the sum of the progression infinitely decreasing. We have: S∞ = a1 / (1 - b). From where we express the denominator: b = 1 - a1 / S∞. It remains to substitute the known values and get the required number: b = 1 - 2 / 3 = -1 / 3 or -0.333(3). We can qualitatively check this result if we remember that for this type of sequence the modulus b should not go beyond 1. As can be seen, |-1 / 3|
Task No. 4. Restoring a series of numbers
Let 2 elements of a number series be given, for example, the 5th is equal to 30 and the 10th is equal to 60. It is necessary to reconstruct the entire series from these data, knowing that it satisfies the properties of a geometric progression.
To solve the problem, you must first write down the corresponding expression for each known term. We have: a5 = b4 * a1 and a10 = b9 * a1. Now divide the second expression by the first, we get: a10 / a5 = b9 * a1 / (b4 * a1) = b5. From here we determine the denominator by taking the fifth root of the ratio of the terms known from the problem statement, b = 1.148698. We substitute the resulting number into one of the expressions for the known element, we get: a1 = a5 / b4 = 30 / (1.148698)4 = 17.2304966.
Thus, we found the denominator of the progression bn, and the geometric progression bn-1 * 17.2304966 = an, where b = 1.148698.
Where are geometric progressions used?
If there were no practical application of this number series, then its study would be reduced to purely theoretical interest. But such an application exists.
Below are the 3 most famous examples:
- Zeno's paradox, in which the nimble Achilles cannot catch up with the slow tortoise, is solved using the concept of an infinitely decreasing sequence of numbers.
- If you place wheat grains on each square of the chessboard so that on the 1st square you put 1 grain, on the 2nd - 2, on the 3rd - 3, and so on, then to fill all the squares of the board you will need 18446744073709551615 grains!
- In the game "Tower of Hanoi", in order to move disks from one rod to another, it is necessary to perform 2n - 1 operations, that is, their number grows exponentially with the number n of disks used.