Sequences-Patterns

A sequence is a set or ordered list of objects or events. A sequence contains numbers or terms which are said
to be the length of the sequence. A set of odd numbers is an example of a sequence.

There are two types of sequences: finite sequences and infinite sequences. A finite sequence is a set of finite numbers.
Example: S = {1, 2, 3……., 10}. An infinite sequence is a sequence in which infinite numbers are presented. Example: a set of even numbers, S = {2, 4, 6, 8……….}.

A pattern may be defined and extended by recursively determining the "next" term. Patterns depend upon the initial condition where the sequence starts and an equation that explains how a term in a sequence can be found from the preceding term.

Example

A sequence between a set of numbers to represent a linear function:

Let us take an example of the given figure in which diagram 1 consists of 3 squares, diagram 2 consists of 5 squares, diagram 3 consists of 7 squares and so on.



If the pattern continues, how many squares will be there in diagram 50 or diagram 100?

Solution

To solve this, we first analyze the number squares in diagram 1 which is equal to 3; then we find the increment of the
squares in the next diagram. In diagram 2, the number of squares increases by 2. We see that as the diagram number
increases, 2 squares in each diagram also increase. If we solve for this then:


No. of diagram

1

2

3

4

5

No. of squares

3

5

7

9

11

Mathematical expression

3+2(0)

3+2(1)

3+2(2)

3+2(3)

3+2(4)

Let us calculate accordingly to find the number of squares in diagram 50 or in diagram 100. As there is an increment of 2
squares with each diagram and the initial diagram has 3 squares, we construct a linear equation to solve this.

We know that the basic form of a linear equation is f (x) = mx + c.

If we compare this expression from the table above, then we find that here

c =3, m = 2 and x= 0, 1, 2, 3, 4…….

The value of x depends upon the diagram number in such a way that if the diagram number is n then the value of x is equal to (n-1).

  • uf (n-1) = 3 + 2(n-1) uwhereu n = diagram number.u

Hence, the number of squares in diagram 50 = 3 + 2(50-1)

= 3 + 98

= 101 squares

And the number of squares in diagram 100 = 3 + 2(100-1).

= 3 + 198

= 201 squares

General Formula for Arithmetic Sequences between a Set of Numbers

As discussed in the example above, if we take the first term as a1 and the common difference as d, then the second
term is a1 + d, the third term is a1 + 2d and so on……..and the nth term is a1 + (n-1) d.

an = a1 + d (n-1)

where an = the last term of the sequence

a1 = the first term of the sequence

n = the number of terms

d = the common difference between the two successive terms

Here's the simpler formula:

d = a2 - a1

= a3 - a2

General form:

D = an – an-1

Hence, the arithmetic series is an example of a linear function in which the dependent quantity is an and the independent
quantity is n.

Example

Find the 10th term of the arithmetic sequence -10, -5, 0, 5 ….

Solution

We know that

an = a1 + d (n-1)

d = -5-(-10) = 5

n =10

a1 = -10

According to formula, a10 = -10 + 5(10-1).

a10 = -10 + 5*9

a10 = -10 + 45

a10 = 35

Try these questions

  1. What is the next term in the given pattern?

    5, 11, 17, 23, ___

    (a) 25

    (b) 29

    (c) 34

    (d) 28

    Answer: B

    a = 5    d = 6

    a5 = 23 + 6 = 29

  2. The first five terms of a linear sequence where a = -6 and d = 4 are

    (a) -6, -2, 2, 6, 10

    (b) -2, 2, 6, 10, 14

    (c) -6, 0, 6, 12, 18

    (d) 4, 10, 16, 22, 28

    Answer: A

    a1 = -6

    a2 = -6 + 4 = -2

    a3 = -2 + 4 = 2

    a4 = 2 + 4 = 6

    a5 = 6 + 4 = 10

  3. What is the 11th term of the given sequence?

    2, 5, 8, 11……

    (a) 32

    (b) 24

    (c) 11

    (d) 39

    Answer: A

    a = 2 d = 3

    an = a + (n-1) d

    a11 = 32

  4. Find the linear sequence where an = 9n – 1.

    (a) 0, 8, 17…….

    (b) 9, 8, 7……..

    (c) 8, 17, 26…….

    (d) 8, 19, 28………

    Answer: C

    an = 9n – 1

    a1 = 9 – 1 = 8

    a2 = 9*2 – 1 = 17

    a3 = 9*3 – 1 = 26

    a4 = 9*4 – 1 = 35

    Hence the sequence is 8, 17, 26, 35…….

  5. Discuss in detail the difference between a finite and an infinite sequence with the help of an example.

    Answer

    When the number of objects in a sequence are finite, i.e. when they are countable, then it is called a Finite Sequence. Let us consider a sequence: 2, 3, 4, 5, 6, 7. In the given series, we know that the count of the number of terms is 6. Hence, it is a finite sequence. Let us take another sequence: 5, 8, 11, …, 65. In this example, we can see that THE last term is present; hence it is also a finite sequence. So, if the last term of a sequence is known or we count the number of terms in that sequence, then it is a finite sequence.

    When the number of objects present in a sequence are infinite, then the sequence is called an Infinite Sequence.
    Let us take an example of such a kind of a sequence: 4, 8, 12… In this example, we cannot count the number of terms as we do not know the last term of the series; hence, it is an infinite sequence. An infinite sequence is very useful for studying vector space, complex analysis, etc.