In the previous section, we completed a discussion on binomial theorem. In this chapter, we will see sequences and series.
The word sequence can be explained using three examples:
Example 1:
This can be written in 4 steps:
1. Consider a bundle of freshly printed currency notes.
2. Let there be a hundred notes in the bundle.
• Each note in that bundle will have a unique number.
3. Suppose that, the 14th note from the top has the number 175214
♦ Then the 15th note from the top will have the number 175215
♦ The 16th note from the top will have the number 175216
♦ so on . . .
4. We say that, the currency notes in the bundle are arranged in a sequential order.
• The sequence can be written as:
175201, 175202, 175203, . . . , 175214, 175215, 175216, 175217, . . . , 175299, 175300.
Example 2
This can be written in 3 steps:
1. Suppose that, we want to make equilateral triangles with identical spheres. Then the arrangement will be as shown in fig.9.1 below:
 |
| Fig.9.1 |
2. The first triangle will contain 3 spheres
♦ The second triangle will contain 6 spheres
♦ The third triangle will contain 10 spheres
♦ The fourth triangle will contain 15 spheres
♦ so on . . .
3. This is the only way to form equilateral triangles using identical spheres.
• So the number of spheres in the triangles form a sequence.
• The sequence can be written as: 3, 6, 10, 15, . . .
Example 3
• Suppose that in a society, the generation gap is 30 years. That is., a person will have children when he/she is 30 years old.
• We want to find the total number of members of a family in each generation for 300 years.
• This can be calculated in 13 steps:
1. Let the total number of members in 300 years be x.
2. Consider a person whose age at present is 30.
• Then 30 years ago, he was not born. His family had upto his father and mother only. So 30 years ago, the number of members will be (x-1)
3. 60 years ago, his father and mother were not born.
• So 60 years ago, the number of members would be [(x-1)-2] = (x-3)
4. 90 years ago, the parents of the persons deducted in (3) were not born.
• Two persons were deducted in (3).
• There would be four parents for those two persons.
• So 90 years ago, the number of members would be [(x-3)-4] = (x-7)
5. 120 years ago, the parents of the persons deducted in (4) were not born.
• 4 persons were deducted in (4).
• There would be 8 parents for those 4 persons.
• So 120 years ago, the number of members would be [(x-7)-8] = (x-15)
6. 150 years ago, the parents of the persons deducted in (5) were not born.
• 8 persons were deducted in (5).
• There would be 16 parents for those 8 persons.
• So 150 years ago, the number of members would be [(x-15)-16] = (x-31)
7. 180 years ago, the parents of the persons deducted in (6) were not born.
• 16 persons were deducted in (5).
• There would be 32 parents for those 16 persons.
• So 180 years ago, the number of members would be [(x-31)-32] = (x-63)
8. 210 years ago, the parents of the persons deducted in (7) were not born.
• 32 persons were deducted in (7).
• There would be 64 parents for those 32 persons.
• So 210 years ago, the number of members would be [(x-63)-64] = (x-127)
9. 240 years ago, the parents of the persons deducted in (8) were not born.
• 64 persons were deducted in (8).
• There would be 128 parents for those 64 persons.
• So 240 years ago, the number of members would be [(x-127)-128] = (x-255)
10. 270 years ago, the parents of the persons deducted in (9) were not born.
• 128 persons were deducted in (9).
• There would be 256 parents for those 128 persons.
• So 270 years ago, the number of members would be [(x-255)-256] = (x-511)
11. 300 years ago, the parents of the persons deducted in (10) were not born.
• 256 persons were deducted in (10).
• There would be 512 parents for those 256 persons.
• So 300 years ago, the number of members would be [(x-511)-512] = (x-1023)
12. But 300 years ago, there will be just two members. A father and mother.
• So we can write: x-1023 = 2
• So x = 1025
• That is., total numbers of members in 300 years = 1025
• We can write: The present person under consideration has 1024 ancestors.
13. Now consider the number of parents, grand parents, great grand parents etc.,
• We get:
♦ Number of parents = 2
♦ Number of grand parents = 4
♦ Number of great grand parents = 8
♦ so on . . .
• These numbers form a sequence. The sequence is:
2, 4, 8, 16, 32, . . .
• Let us see some features related to sequences. They can be written in steps:
1. The various numbers occurring in a sequence are called terms of the sequence.
• For example, we can write:
‘8’ is a term of the sequence 2, 4, 8, 16, 32, . . .
2. The terms are denoted by the letter a.
• A subscript is also given to ‘a’. The subscript will indicate the position of the term.
• For example, we can write:
In the sequence 2, 4, 8, 16, 32, . . . , a4 = 16
3. an denotes the term at the nth position in the sequence. This term is also called the general term of the sequence.
4. If the number of terms in a sequence is finite, then that sequence is called a finite sequence.
• For example, the sequence in example 1 that we saw above, is a finite sequence. This is because, the number of terms in that sequence is 100, which is a finite number. (Something that is finite has a definite fixed size or extent)
• The sequence in example 3 is also finite because, we are considering the number of generations within 300 years.
5. If the number of terms in a sequence is infinite, then that sequence is called a infinite sequence.
• For example, the sequence in example 2 that we saw above, is an infinite sequence. This is because, infinite number of such triangles are possible and correspondingly, infinite number of terms will be present in that sequence. (Something that is infinite, is limitless or endless in space, extent, or size. It is impossible to count, measure or calculate an infinite quantity)
Algebraic formula for the nth term of a sequence
This can be explained with the help of some examples.
Example 1:
This can be written in 4 steps:
1. Consider the sequence 2, 4, 6, 8, . . .
• It is the sequence of even natural numbers.
2. We have:
• a1 = 2
♦ 2 can be obtained by multiplying the ‘position number 1’ by 2
• a2 = 4
♦ 4 can be obtained by multiplying the ‘position number 2’ by 2
• a3 = 6
♦ 6 can be obtained by multiplying the ‘position number 3’ by 2
• a4 = 8
♦ 8 can be obtained by multiplying the ‘position number 4’ by 2
so on . . .
3. So it is clear that, any term in this sequence can be obtained by multiplying it’s ‘position number’ by 2
That is., an = n × 2
4. We can write:
The algebraic formula for the nth term of the sequence 2, 4, 6, 8, . . . is: an = 2n
Example 2:
This can be written in 4 steps:
1. Consider the sequence 1, 3, 5, 7, . . .
• It is the sequence of odd natural numbers.
2. We have:
• a1 = 1
♦ 1 can be obtained by multiplying the ‘position number 1’ by 2 and then subtracting 1
♦ That is., 1 = (1 × 2) - 1
• a2 = 3
♦ 3 can be obtained by multiplying the ‘position number 2’ by 2 and then subtracting 1
♦ That is., 3 = (2 × 2) - 1
• a3 = 5
♦ 1 can be obtained by multiplying the ‘position number 3’ by 2 and then subtracting 1
♦ That is., 5 = (3 × 2) - 1
• a4 = 7
♦ 7 can be obtained by multiplying the ‘position number 4’ by 2 and then subtracting 1
♦ That is., 7 = (4 × 2) - 1
so on . . .
3. So it is clear that, any term in this sequence can be obtained by multiplying it’s ‘position number’ by 2 and then subtracting 1
That is., an = n × 2 - 1
4. We can write:
The algebraic formula for the nth term of the sequence 1, 3, 5, 7, . . . is:
an = 2n - 1
Example 3:
This can be written in 5 steps:
1. Consider the sequence 1, 1, 2, 3, 5, 8, . . .
• Here we do not see a visible pattern.
2. We have:
• a1 = 1
• a2 = 1
• a3 = 2
♦ 2 can be obtained by adding a2 and a1.
♦ a2 and a1 are the two terms coming just before a3
• a4 = 3
♦ 3 can be obtained by adding a3 and a2.
♦ a3 and a2 are the two terms coming just before a4.
• a5 = 5
♦ 5 can be obtained by adding a4 and a3.
♦ a4 and a3 are the two terms coming just before a5.
• a6 = 8
♦ 8 can be obtained by adding a5 and a4.
♦ a5 and a4 are the two terms coming just before a6.
so on . . .
3.
So it is clear that, any term in this sequence can be obtained by
adding the two terms coming just before it.
That is., an = an-1 + an-2.
4. We can write:
The algebraic formula for the nth term of the sequence 1, 1, 2, 3, 5, 8, . . . is:
an = an-1 + an-2, n > 2
• Note that, n must be greater than 2. If we put n = 2, then an-2 will denote the zeroth term, which is not available.
5. This sequence is called Fibonacci sequence.
• Now we have a basic idea about the algebraic formula for the nth term.
• In the formula, we put natural numbers starting from 1 on the left side. We get corresponding values on the right side.
• Based on this information, we can consider a sequence as a function of the form a(n)
• In some cases, the domain will be the set of natural numbers. {1, 2, 3, 4, . . .}
♦ Then it will be an infinite sequence.
• In some cases, the domain will be a subset of natural numbers. {1, 2, 3, 4, . . . k}
♦ Then it will be a finite sequence.
• The range will be the terms of the sequence.
• There are some sequences which do not have an algebraic formula for the nth term.
• For example, consider the sequence 2, 3, 5, 7, 11, 13, 17, . . .
• It is the sequence of prime numbers. There is no algebraic formula to obtain the terms of this sequence.
• In such cases, we describe the sequence by a verbal description. This verbal description is called the rule for generating the terms of the sequence.
Series
Basics about series can be written in steps:
1. Let a1, a2, a3, a4, . . . an be a given sequence.
• Then the expression a1 + a2 + a3 + a4 + . . . an + . . . is called the series associated with the given sequence.
2. Note that the expression a1 + a2 + a3 + a4 + . . . an + . . . is the series.
• We must not consider the actual sum as the series.
• The actual sum is referred to as sum of the series.
3. We can write about finite and infinite series:
• If the given sequence is finite, then the associated series is a finite series.
• If the given sequence is infinite, then the associated series is an infinite series.
4. Since the series involves a summation, we can use the sigma notation. We have seen the details about sigma notation in a previous chapter. [see section 8.1]
• So the series can be abbreviated as: $\sum\limits_{k\,=\,1}^{k\,=\,n}{a_k}$
•
'k=1' at the bottom of '∑' indicates that, the value of k starts from
1. In other words, the value of k in the first term is 1.
• 'k=n' at the top of '∑' indicates that, the value of k ends at
n. In other words, the value of k in the last term is n.
• So the values of k are: 1, 2, 3, . . . n
• It is known as the summation from k = 1 to k = n.
• Let us write an example:
The series associated with the sequence of odd natural numbers can be written in two forms:
(i) 1 + 3 + 5 + 7 + . . .
(ii) $\sum\limits_{k\,=\,1}^{k\,=\,\infty}{2k-1}$
In the next section we will see some solved examples on sequences and series.
Copyright©2022 Higher secondary mathematics.blogspot.com
No comments:
Post a Comment