Information Processing

Introduction:

In general days, we may observe patterns in both nature and man-made things. In nature, patterns seem in tress, leaves, movements in celestial bodies and many others. In man-made. Patterns occurs in structures, buildings, and many more. It is really wonder to know all beautiful patterns are based on mathematics.

We have to find the relationship between the different types of patterns and structures; E.g. patterns in fabric designs,

Tables and patterns leading to linear functions

Situation 1: Observe the pattern carefully,

Consider the circular disc, now observe the pattern of shapes in the tabular form;

Number of steps (x)	1	2	3	4	…..
Number of circular ring (y)	1	3	5	7	……

Here, you find the relationship between the number of steps and number of circular rings in each step. So, the relation between the variable assumed x and y can be generalised as y=2x-1.

Example 1: In the following figures, polygons are formed by increasing the number of sides using matchsticks as given below.

Find the number of sticks required to form the next three shapes by tabulation and generalisation?

Solution:

In the above pattern of polygons, in the first shape (x = 1), we get a closed shape called a triangle. Similarly, the second shape (x = 2) gives a four-sided polygon and the third shape (x = 3) is a five-sided polygon and continuing in the same way two more shapes are formed. If the number of match sticks required to form each of the shapes is taken as y, then the values of x and y are tabulated as given below.

X	1	2	3	4	5	…..
y	3	4	5	6	7	……

Observe the table given above;

when x = 1, y = 3 = 1+2

when x = 2, y = 4 = 2+2

when x = 3, y = 5 = 3+2

when x = 4, y = 6 = 4+2

when x = 5, y = 7 = 5+2

Hence, each of the values of y which we get from the table is 2 more than x. That is y x = + 2.

Therefore, 6th shape (x=6) will have y = 8 = 6+2 (8 match sticks) and so on.

Some Problems:

1. Write the appropriate number of pattern and its generalisation:

(i)

(ii)

2. Identify the correct relationship between x and y from the given table

X	1	2	3	4	……
Y	4	8	12	16	……

(i) Y=4x

(ii) y =x + 4

(iii) y = 4

(iv) y =4 * 4

3. Identify the correct relationship between x and y from the given table

X	-2	-1	0	1	2	……
Y	6	3	0	-3	-6	……

(i) y =x−2

(ii) y = x+2

(iii) y= x +3

(iv) y = -3x

Pascal’s triangle

It is a triangular array of the binomial coefficients. Write a function that takes an integer value n as input and prints first n lines of the Pascal’s triangle. Following are the first 6 rows of Pascal’s Triangle.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

I.e. also,

Example: Complete the following Pascal’s Triangle by observing the number pattern.

Solution:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Example: Observe the sequence of numbers obtained in the 3rd and 4th slanting rows of Pascal’s Triangle and find the difference between the consecutive numbers and complete the table given below.

(i)

3^rd slanting row	1	3	6	10	15	21
difference		2	------	4	-----	6

Sol: 2, 3, 4, 10, 6

(ii)

4^th slanting row	1	4	10	20	35
Difference		3	------	10	------

Sol: 3, 6, 10, 15.

Example: Tabulate the 3rd slanting row of the Pascal’s Triangle by taking the position of the numbers in the slanting row as x and the corresponding values as y.