MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: R

There are no items in this cart.
Continue Shopping
Menu
Get instant 20% OFF on Online Material.
coupon code: MOB20 | View Course list

Get extra R 220 off
USE CODE: omar10

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.

Sequence

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
Representation of a SequenceThus, if we are writing the third term it is written as x3. If we have a sequence of the type {3, 5, 7, ….} then we can represent this sequence by the rule xn = 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 xn = 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

  an = 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 xn = 2n.

Hence, in general, an expression of the form {a, ar, ar2, ….. } 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

  an = 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. a2, b2, c2 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.

askIITians offers extensive study material which covers all the important topics of JEE. It covers all the basic concepts of series and sequences which fetches several questions in the JEE. It is important to have a good hold on all the topics in order to remain competitive in the JEE.

Related resources:

To read more, Buy study materials of Sequences and Series comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.

  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: R 15,000
  • View Details
Get extra R 1,500 off
USE CODE: omar10