# Patterns in Numbers

We see so many patterns around us in our daily life. We know that a pattern is an arrangement of objects, colors, or numbers placed in a certain order. Some patterns neither grow nor reduce but only repeat. Such patterns are known as repeating patterns. A pattern has a group of units that follow a rule while repeating or changing. Some examples of patterns are given below:

## Patterns in Whole Numbers

Whole numbers can be represented by line segments, triangles, squares, rectangles, etc.

1. Every whole number greater than 1 can be arranged in a line as shown below:

2. Some whole numbers can be represented by triangles. Such numbers are called triangular numbers.

3. Some whole numbers can be represented by squares. These numbers are also known as perfect squares.

4. Some whole numbers can be represented as rectangles.

 3 Consecutive Numbers 5 Consecutive Numbers 1 + 2 + 3 = 62 + 3 + 4 = 93 + 4 + 5 = 124 + 5 + 6 = 155 + 6 + 7 = 18 1 + 2 + 3 + 4 + 5 = 152 + 3 + 4 + 5 + 6 = 203 + 4 + 5 + 6 + 7 = 254 + 5 + 6 + 7 + 8 = 305 + 6 + 7 + 8 + 9 = 356 + 7 + 8 + 9 + 10 = 40 Sums are the multiples of 3 Sums are the multiples of 5

Let us have a quick review of what we have learnt earlier about patterns.

I. Complete the series by drawing the next figure:

A pattern can also be created with numbers. The set of numbers which follow a common rule form a pattern. For example, the sequence 2, 4, 6, 8, …… can be extended by using the rule of even numbers. Patterns with numbers can also be created using mathematical operations like addition, subtraction, multiplication and division.

For example:

1. Write the next 3 terms of the pattern 11, 15, 19, 23, ……..

The first term is 11. The common difference is 4. The next 3 terms are 27, 31, 35.

## Pattern Observation

The mathematical calculations can be simplified by observing certain patterns.

1. Addition of 9, 99, 999, 9999, etc. to a whole number

For Example:

 115 + 9115 + 99115 + 999115 + 9999115 + 99999 = 115 + 10 - 1= 115 + 100 - 1= 115 + 1000 - 1= 115 + 10000 - 1= 115 + 100000 - 1 = 125 - 1= 215 - 1 = 1115 - 1= 10115 - 1= 100115 - 1 = 124= 214= 1114= 10114= 100114

2. Subtraction of 9, 99, 999, etc. from a whole number.

For Example:

 2345 - 92345 - 992345 - 999 = 2345 - (10 - 1)= 2345 - (100 - 1)= 2345 - (1000 - 1) = 2345 - 10 + 1= 2345 - 100 + 1= 2345 - 1000 + 1 = 2335 + 1 = 2336= 2245 + 1 = 2246= 1345 + 1 = 1346

3. Multiplication of a whole number by 9, 99, 999, etc.

For Example:

 125 × 9125 × 99125 × 999 = 125 (10 - 1)= 125 (100 - 1)= 125 (1000 - 1) = 1250 - 125= 12500 - 125= 125000 - 125 = 1125= 12375= 124875

Here are some patterns. Follow them and extend them:

(i)

1 × 9 + 2 = 11

12 × 9 + 3 = 111

123 × 9 + 4 = 1111

1234 × 9 + 5 = 11111

12345 × 9 + 6 = _______

123456 × 9 + 7 = _______

(ii)

1 × 8 + 1 = 9

12 × 8 + 2 = 98

123 × 8 + 3 = 987

1234 × 8 + 4 = 9876

12345 × 8 + 5 = _______

123456 × 8 + 6 = _______

(iii)

111 ÷ 3 = 37

222 ÷ 6 = 37

333 ÷ 9 = 37

444 ÷ 12 = 37

555 ÷ 15 = _______

666 ÷ 18 = _______

(iv)

9 + 1 = 10

90 + 10 = 100

900 + 100 = 1000

9000 + 1000 = _______

90000 + 10000 = _______

900000 + 100000 = _______

Follow the pattern of the following products and extend them further:

(i)

5 × 5 = 25

55 × 5 = 275

555 × 5 = 2775

5555 × 5 = 27775

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

55555555 × 5 = 277777775

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

(ii)

404 × 404 = 163216

505 × 505 = 255025

606 × 606 = 367236

707 × 707 = 499849

808 × 808 = 652864

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _

(iii) Study the following pattern and write the next two steps.

× 1 = 1

11 × 11 = 121

111 × 111 = 12321

1111 × 1111 = 1234321

____ × _____ = ________

_____ × _____ = _________

The next two steps will be

11111 × 11111 = 123454321

111111 × 111111 = 12345654321

Worksheet on Patterns in Whole Numbers:

In patterns in numbers here are some unsolved questions for the students to understand and complete the questions on patterns.

I. Complete the given series:

(i) 1, 3, 5, 7, …….., …….., …….., ……..

(ii) 5, 10, 15, 20, …….., …….., …….., ……..

(iii) 10, 20, 30, 40, …….., …….., …….., ……..

(iv) 3, 5, 8, 12, 17, …….., …….., …….., ……..

(v) 10, 20, 35, 55, 80, …….., …….., …….., ……..

(vi) 2, 2, 3, 3, 3, 4, 4, 4, 4, …….., …….., …….., ……..

(vii) 1, 3, 6, 8, 11, 13, 16, …….., …….., …….., ……..

(viii) 35, 45, …….., 65, 75, …….., 95

(ix) 100, 90, 80, …….., 60, 50 ……..

(x) 1, 3, 5, 7, 9, …….., …….., …….., ……..

I. (i) 9, 11, 13, 15

(ii) 25, 30, 35, 40

(iii) 50, 60, 70, 80

(iv) 23, 30, 38, 47

(v) 110, 145, 185, 230

(vi) 5, 5, 5, 5, 5

(vii) 18, 21, 23, 26

(viii) 55, 85

(ix) 70, 40

(x) 11, 13, 15, 17

II. Complete the series:

(i) 1, 4, 7, 10, 13, 16, …….., …….., …….., ……..

(ii) 12, 17, 22, 27, 32, …….., …….., …….., ……..

(iii) 6, 16, 26, 36, 46, …….., …….., …….., ……..

(iv) 10, 30, 50, 70, …….., …….., …….., ……..

(v) 38, 35, 32, 29, …….., …….., …….., ……..

(vi) 54, 50, 46, 42, …….., …….., …….., ……..

(vii) 53, 49, 45, 41, …….., …….., …….., ……..

(viii) 23, 30, 37, 44, …….., …….., …….., ……..

II. (i) 19, 22, 25, 28

(ii) 137, 42, 47, 52

(iii) 56, 66, 76, 86

(iv) 90, 110, 130, 150

(v) 26, 23, 20, 17

(vi) 38, 34, 30, 26

(vii) 37, 33, 29, 25

(viii) 51, 58, 65, 72

III. Complete the following series:

(i) 7, 14, 21, 28, …….., …….., …….., ……..

(ii) 25, …….., 75, 100, …….., …….., …….., ……..

(iii) 10, 100, …….., 10000, 100000, ……..

(iv) 2, 10, …….., 250, 1250

(v) 500000, 50000, …….., 500, …….., 5

(vi) 64, 32, 16, …….., 4, …….., 1

(vii) 5, 10, 20, …….., 80, …….., 320

(viii) 8, 16, 24, 32, …….., …….., …….., ……..

(ix) 45, 54, 63, …….., …….., …….., ……..

(x) 2, 6, 18, 54, ……..

III. (i) 7, 14, 21, 28, …….., …….., …….., ……..

(ii) 25, …….., 75, 100, …….., …….., …….., ……..

(iii) 10, 100, …….., 10000, 100000, ……..

(iv) 2, 10, …….., 250, 1250

(v) 500000, 50000, …….., 500, …….., 5

(vi) 64, 32, 16, …….., 4, …….., 1

(vii) 5, 10, 20, …….., 80, …….., 320

(viii) 8, 16, 24, 32, …….., …….., …….., ……..

(ix) 45, 54, 63, …….., …….., …….., ……..

(x) 2, 6, 18, 54, ……..

IV. Using the shorter method simplify the following.

(i) 226 + 99

(ii) 2157 + 9999

(iii) 1239 + 99999

(iv) 6712 - 999

(v) 42785 - 9999

(vi) 1235 × 999

IV. (i) 226 + 99

= 226 + 100 - 1

= 326 - 1

=  325

(ii) 2157 + 9,999

= 2,157 + 10,000 - 1

= 12,157 - 1

= 12,156

(iii) 1239 + 99999

= 1239 + 100000 - 1

= 101239 - 1

= 101238

(iv) 6712 - 999

= 6712 - (1000 - 1)

= 6712 - 1000 + 1

= 5712 + 1

= 5713

(v) 42785 - 9999

= 42785 - (10,000 - 1)

= 42785 - 10,000 + 1

= 32785 + 1

= 32786

(vi) 1235 × 999

= 1235 × (1000 - 1)

= 123500 - 1235

= 1233765

2. Observe the pattern and fill in the blanks.

1 × 9 + 1 = 10

12 × 9 + 2 = 110

123 × 9 + 3 = 1110

1234 × 9 + 4 = _____

12345 × 9 + 5 = ______

2.

1234 × 9 + 4 = 11110

12345 × 9 + 5 = 111110

3. Study the following pattern and write the next two steps.

1 × 8 + 1 = 9

12 × 8 + 2 = 98

123 × 8 + 3 = 987

1234 × 8 + 4 = 9876

12345 × 8 + 5 = 98765

______ × _ + _ = ______

_______ × __ + __ = _______

3.

123456 × 8 + 6 = 987654

1234567 × 8 + 7 = 9876543

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