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Basic Concepts of Sequences and Pattern Sequence A sequence is a set of values which are in a particular order. In simple words, we can say that a succession of numbers formed according to some definite rule is called a sequence. Mathematically, a sequence is defined as a map whose domain is the set of natural numbers (which may be finite or infinite) and the range may be the set of real numbers or complex numbers. We can represent it as f: N → X, where N is the set of naturals. The sequence is finite or infinite depending on whether the set ‘N’ is finite or infinite. If ‘X’ is the set of real numbers then f is said to be a real valued sequence while if the set X is of complex numbers, then f is termed as a sequence of complex numbers. Difference between sequence and set…. A sequence is often confused with a set. Though they both appear to be same yet they are different. A sequence is almost the same as a set except for the fact that in a set the elements cannot repeat while it is not so in case of a sequence. Moreover, there is no importance of order in a set, while the order matters a lot in sequence. The following example will further clear the difference between the two: {2, 4, 2, 4, ….} is an alternating sequence of 2s and 4s. The corresponding set would be just {2, 4}. Representation of a Sequence: A sequence is generally represented by placing a subscript next to x which denotes the term number in the sequence. Mathematically, we write it as Thus, if we are writing the third term it is written as x_{3}. If we have a sequence of the type {3, 5, 7, ….} then we can represent this sequence by the rule x_{n} = 2n +1. Sequence and pattern If the terms of a sequence follow a particular pattern or a rule, then such a sequence is termed as a progression. Basically, there are three types of progressions: (i) Arithmetic Progression (A.P.) (ii) Geometric Progression (G.P.) (iii) Harmonic Progression (H.P.) We shall just give an outline of the three kinds of sequences as they have been discussed in detail in the coming sections. 1. Arithmetic Progression: If the difference between two consecutive terms in a sequence is constant then the sequence is termed as an arithmetic sequence or an arithmetic progression. For example {1, 4, 7, 10, …. } is a sequence in which every term has a difference of 3. Hence, such a sequence is called as an arithmetic progression. We can denote this sequence by the rule x_{n }= 3n -2. Hence, in general, an expression of the form {a, a+d, a+2d, ….. } is called as an arithmetic progression. Here, a is called the first term of the sequence while d is called the common difference. The nth term of an A.P. is given by
Sequence
A sequence is a set of values which are in a particular order. In simple words, we can say that a succession of numbers formed according to some definite rule is called a sequence.
Mathematically, a sequence is defined as a map whose domain is the set of natural numbers (which may be finite or infinite) and the range may be the set of real numbers or complex numbers. We can represent it as
f: N → X, where N is the set of naturals. The sequence is finite or infinite depending on whether the set ‘N’ is finite or infinite. If ‘X’ is the set of real numbers then f is said to be a real valued sequence while if the set X is of complex numbers, then f is termed as a sequence of complex numbers.
Difference between sequence and set….
A sequence is often confused with a set. Though they both appear to be same yet they are different. A sequence is almost the same as a set except for the fact that in a set the elements cannot repeat while it is not so in case of a sequence. Moreover, there is no importance of order in a set, while the order matters a lot in sequence. The following example will further clear the difference between the two:
{2, 4, 2, 4, ….} is an alternating sequence of 2s and 4s.
The corresponding set would be just {2, 4}.
Representation of a Sequence:
A sequence is generally represented by placing a subscript next to x which denotes the term number in the sequence. Mathematically, we write it as Thus, if we are writing the third term it is written as x_{3}. If we have a sequence of the type {3, 5, 7, ….} then we can represent this sequence by the rule x_{n} = 2n +1.
Sequence and pattern
If the terms of a sequence follow a particular pattern or a rule, then such a sequence is termed as a progression. Basically, there are three types of progressions:
(i) Arithmetic Progression (A.P.)
(ii) Geometric Progression (G.P.)
(iii) Harmonic Progression (H.P.)
We shall just give an outline of the three kinds of sequences as they have been discussed in detail in the coming sections.
1. Arithmetic Progression: If the difference between two consecutive terms in a sequence is constant then the sequence is termed as an arithmetic sequence or an arithmetic progression. For example {1, 4, 7, 10, …. } is a sequence in which every term has a difference of 3. Hence, such a sequence is called as an arithmetic progression. We can denote this sequence by the rule x_{n }= 3n -2.
Hence, in general, an expression of the form {a, a+d, a+2d, ….. } is called as an arithmetic progression.
Here, a is called the first term of the sequence while d is called the common difference. The nth term of an A.P. is given by
a_{n} = a+(n-1)d
View the video for more on arithmetic progression
2. Geometric Progression: If the terms of a sequence are such that that they can be obtained by multiplying the previous term by a particular number then it called to be a geometric sequence. For example {2, 4, 8, 16, …. } is a sequence in which every term is obtained by multiplying the previous term by 2. Hence, such a sequence is called as a geometric progression. We can denote this sequence by the rule x_{n }= 2^{n}.
Hence, in general, an expression of the form {a, ar, ar^{2}, ….. } is called a geometric progression.
Here, a is called the first term of the sequence while r is called the common ratio. The nth term of a G.P. is given by
a_{n} = ar^{(n-1)}
3. Harmonic Progression: If we have a sequence and if the reciprocals of the terms of the sequence form an arithmetic progression then the sequence is termed to be in Harmonic Progression abbreviated as H.P.
A general H.P. is 1/a + 1/(a + d) + 1/(a + 2d) + ......
The general term of the H.P. is given by
1/ [a + (n -1) d].
Illustration: If ln (a+c), ln (c-a), ln (a-2b+c) are in A.P. then
1. a, b, c are in A.P 3. a^{2}, b^{2}, c^{2 }are in A.P
2. a, b, c are in G.P 4. a, b, c are in H.P
Solution: It is given in the question that ln (a+c), ln (c-a), ln (a-2b+c) are in A.P
Hence, this implies that (a+c), (c-a), (a-2b+c) are in G.P
So, (c-a)^{2} = (a+c)(a-2b+c)
(c-a)^{2} = (a+c)^{2}- 2b(a+c)
2b(a+c) = (a+c)^{2} – (c-a)^{2}
2b(a+c) = 4ac
Hence b = 2ac/(a+c) which means that a, b and c are in H.P.
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