Number Patterns and Sequences

Mathematics

Class 5 | CBSE, NCERT & ICSE

Learning Objectives

Identify and extend number patterns (growing, shrinking, repeating)
Find the rule behind a given number sequence
Understand arithmetic patterns (constant addition or subtraction)
Understand geometric patterns (constant multiplication or division)
Explore the Fibonacci sequence and its occurrence in nature
Work with triangular numbers, square numbers, and cube numbers
Create number patterns using given rules
Apply pattern recognition to solve problems and make predictions
Identify patterns in multiplication tables, Pascal's triangle, and magic squares

Introduction — Patterns Are Everywhere!

Look around you — patterns are everywhere! The petals of a flower, the stripes on a tiger, the tiles on a floor, the rhythm of music, and even the way numbers behave. In mathematics, a pattern is a set of numbers or shapes that follow a definite rule.

A sequence is a list of numbers written in a specific order, where each number follows a rule. For example: 2, 4, 6, 8, 10, ... — can you see the pattern? Each number is 2 more than the previous one. Finding patterns helps us predict what comes next, and this skill is used in science, computer programming, music, and art!

Image needed: A collage showing patterns in nature and maths — sunflower spirals, Rangoli geometric patterns, number sequences on a chalkboard, and a peacock feather. Connecting nature to maths.
Style: Collage-style illustration, vibrant colours, Indian elements (Rangoli, peacock)
Size: 700×400px

Types of Number Patterns

1. Arithmetic Patterns (Add or Subtract a Constant)

In an arithmetic pattern, you add (or subtract) the same number each time. This constant is called the common difference.

Pattern	Rule	Common Difference	Next 3 Terms
5, 10, 15, 20, ...	Add 5	+5	25, 30, 35
3, 7, 11, 15, ...	Add 4	+4	19, 23, 27
100, 92, 84, 76, ...	Subtract 8	−8	68, 60, 52
1.5, 3.0, 4.5, 6.0, ...	Add 1.5	+1.5	7.5, 9.0, 10.5

Example 1: Find the next 3 terms: 7, 12, 17, 22, ...

Solution: Common difference = 12 − 7 = 5 (adding 5 each time)

Next terms: 22 + 5 = 27, 27 + 5 = 32, 32 + 5 = 37

Answer: 7, 12, 17, 22, 27, 32, 37

Example 2: Find the rule and the missing number: 95, 86, 77, ___, 59, 50

Solution: Difference = 86 − 95 = −9 (subtracting 9 each time)

Missing number = 77 − 9 = 68

2. Geometric Patterns (Multiply or Divide by a Constant)

In a geometric pattern, you multiply (or divide) by the same number each time. This constant is called the common ratio.

Pattern	Rule	Common Ratio	Next 3 Terms
2, 6, 18, 54, ...	Multiply by 3	×3	162, 486, 1458
5, 10, 20, 40, ...	Multiply by 2	×2	80, 160, 320
10000, 1000, 100, 10, ...	Divide by 10	÷10	1, 0.1, 0.01
3, 12, 48, 192, ...	Multiply by 4	×4	768, 3072, 12288

Example 3: Find the next 2 terms: 4, 12, 36, 108, ...

Solution: 12 ÷ 4 = 3, 36 ÷ 12 = 3, 108 ÷ 36 = 3 → Common ratio = ×3

Next: 108 × 3 = 324, 324 × 3 = 972

Answer: ..., 108, 324, 972

3. Repeating Patterns

A group of numbers repeats over and over.

Example 4: 1, 4, 9, 1, 4, 9, 1, 4, 9, ...

The group {1, 4, 9} repeats. The 10th term = 1 (start of 4th cycle).

Special Number Sequences

Square Numbers

A square number is the result of multiplying a number by itself: n × n = n²

n	1	2	3	4	5	6	7	8	9	10
n²	1	4	9	16	25	36	49	64	81	100

Pattern in square numbers: The difference between consecutive squares is always odd: 4−1=3, 9−4=5, 16−9=7, 25−16=9, ... (differences: 3, 5, 7, 9, 11, ...)

Triangular Numbers

Triangular numbers count dots arranged in triangles: 1, 3, 6, 10, 15, 21, 28, ...

Rule: Add 2, then 3, then 4, then 5, ... (add one more each time)

Position	1st	2nd	3rd	4th	5th	6th	7th	8th
Number	1	3	6	10	15	21	28	36
How	1	1+2	1+2+3	1+2+3+4	1+2+3+4+5	+6	+7	+8

Image needed: Visual representation of triangular numbers using dots arranged in triangles — showing 1 dot, 3 dots (triangle), 6 dots (bigger triangle), 10 dots, 15 dots. Each triangle clearly labelled with its number.
Style: Colourful dot diagrams, clean and educational
Size: 650×300px

Cube Numbers

A cube number is n × n × n = n³. They represent the volume of cubes.

n	1	2	3	4	5	6
n³	1	8	27	64	125	216

The Fibonacci Sequence

One of the most famous patterns in mathematics! Each number is the sum of the two numbers before it:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Rule: Next term = Previous two terms added together

0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13 ... and so on

Did You Know? The Fibonacci sequence appears in nature! Sunflowers have 34 or 55 spirals, pinecones have 8 or 13 spirals, and many flowers have 3, 5, 8, or 13 petals. Indian mathematician Virahanka described this sequence around 700 CE — centuries before Fibonacci!

Image needed: The Fibonacci spiral in nature — showing the number sequence, the spiral drawn on squares, and real examples: sunflower head with spiral lines, a shell (nautilus), and a pinecone. Include a small portrait of Virahanka/Indian connection note.
Style: Educational diagram combining the spiral with nature photos/illustrations
Size: 700×450px

How to Find the Rule of a Pattern

Follow these steps:

Step	What to Do	Example: 3, 7, 11, 15, 19, ...
1	Find the differences between consecutive terms	7−3=4, 11−7=4, 15−11=4, 19−15=4
2	If differences are constant → Arithmetic (add/subtract)	Yes! Constant difference = 4
3	If differences are NOT constant, find ratios	Not needed here
4	If ratios are constant → Geometric (multiply/divide)	—
5	If neither, look for other rules (Fibonacci, squares, two-step)	—

Two-Step Rules

Some patterns use two operations together:

Example 5: 2, 5, 11, 23, 47, ...

Solution: Differences: 3, 6, 12, 24 — the differences are doubling!

Or think: each term = previous × 2 + 1

Check: 2×2+1=5 ✓, 5×2+1=11 ✓, 11×2+1=23 ✓, 23×2+1=47 ✓

Next term: 47 × 2 + 1 = 95

Example 6: 1, 4, 9, 16, 25, 36, ...

Solution: These are 1², 2², 3², 4², 5², 6² — square numbers!

Next term: 7² = 49

Example 7: 2, 6, 12, 20, 30, 42, ...

Solution: Differences: 4, 6, 8, 10, 12 — differences increase by 2 each time

Next difference = 14. Next term = 42 + 14 = 56

(These are n × (n+1): 1×2=2, 2×3=6, 3×4=12, 4×5=20, 5×6=30, 6×7=42, 7×8=56)

Patterns in Multiplication Tables

Multiplication tables are full of hidden patterns!

The 9-Times Table Pattern

9 × n	Answer	Tens Digit	Ones Digit	Digit Sum
9 × 1	09	0	9	0+9 = 9
9 × 2	18	1	8	1+8 = 9
9 × 3	27	2	7	2+7 = 9
9 × 4	36	3	6	3+6 = 9
9 × 5	45	4	5	4+5 = 9
9 × 6	54	5	4	5+4 = 9

Patterns: Tens digit goes up (0,1,2,3,...), ones digit goes down (9,8,7,6,...), and digit sum is always 9!

Magic Squares

A magic square is a grid where every row, column, and diagonal adds up to the same number (the magic constant).

3×3 Magic Square (numbers 1-9, magic constant = 15):

2	7	6
9	5	1
4	3	8

Every row, column, and diagonal = 15

Image needed: A colourful 3×3 magic square with arrows showing how rows, columns, and diagonals all sum to 15. Also show a 4×4 magic square (like Khajuraho temple magic square from Indian heritage) with magic constant 34.
Style: Colourful grid with arrows and sums marked, Indian temple context
Size: 600×400px

Did You Know? India has one of the oldest known magic squares! A 4×4 magic square is carved in the Parshvanath temple in Khajuraho (12th century). Its magic constant is 34. Ancient Indian mathematicians were fascinated by number patterns!

Patterns in Nature and Real Life

Where	Pattern	Type
Sunflower seeds	Fibonacci spirals (34, 55)	Fibonacci
Honeycomb	Hexagonal pattern repeats	Repeating geometric
Seasons	Summer, Monsoon, Winter, Spring — repeats	Repeating
Cricket overs	6 balls per over, score increases	Arithmetic
Bank interest (compound)	Amount doubles periodically	Geometric
Rangoli designs	Symmetrical repeating patterns	Repeating / symmetry
Train timings	Arrive every 15 minutes	Arithmetic (+15)

Key Words

Word	Meaning
Pattern	A set of numbers/shapes that follow a definite rule
Sequence	An ordered list of numbers following a rule
Term	Each number in a sequence
Common Difference	The constant amount added/subtracted in arithmetic patterns
Common Ratio	The constant multiplier in geometric patterns
Arithmetic Pattern	Add or subtract the same number each time
Geometric Pattern	Multiply or divide by the same number each time
Fibonacci Sequence	Each term = sum of two previous terms
Square Number	n × n (1, 4, 9, 16, 25, ...)
Triangular Number	1, 3, 6, 10, 15, ... (sum of first n natural numbers)
Magic Square	Grid where all rows, columns, diagonals have the same sum

Key Points to Remember

Arithmetic patterns: add/subtract a constant (common difference)
Geometric patterns: multiply/divide by a constant (common ratio)
Fibonacci: each term = sum of two previous terms (Indian origin — Virahanka, 700 CE)
Square numbers: 1, 4, 9, 16, 25, ... (differences are odd: 3, 5, 7, 9, ...)
Triangular numbers: 1, 3, 6, 10, 15, ... (add 2, 3, 4, 5, ...)
To find the rule: look at differences first, then ratios, then other rules
Some patterns combine two operations (×2 then +1)
The 9-times table: digits always sum to 9
Patterns appear everywhere — nature, art, music, sports, daily life

Finished learning? Now test what you know!

🎯 Quick Quiz — Test Yourself!

Answer these questions right here. You get instant feedback — no paper needed!

Score: 0 / 8

1. What comes next: 5, 10, 15, 20, ___?

Add 5 each time: 20 + 5 = 25.

2. What comes next: 2, 4, 8, 16, ___?

Each term doubles: 16 × 2 = 32.

3. In the pattern 3, 6, 9, 12, … the rule is:

These are multiples of 3 — add 3 each time.

4. Which are the first four square numbers?

Square numbers: 1×1, 2×2, 3×3, 4×4 = 1, 4, 9, 16.

5. What comes next: 6, 12, 18, 24, ___?

Add 6 each time: 24 + 6 = 30.

6. What is the missing number: 5, 10, 15, ___, 25?

The pattern adds 5 each time, so the number between 15 and 25 is 20.

7. Fibonacci pattern: 1, 1, 2, 3, 5, ___ (each number is the sum of the two before it).

3 + 5 = 8.

8. What comes next: 20, 16, 12, 8, ___?

Subtract 4 each time: 8 − 4 = 4.

Practice Questions

A. Find the Next 3 Terms

1. 4, 9, 14, 19, ___, ___, ___

2. 100, 88, 76, 64, ___, ___, ___

3. 3, 6, 12, 24, ___, ___, ___

4. 1, 4, 9, 16, 25, ___, ___, ___

5. 1, 1, 2, 3, 5, 8, ___, ___, ___

6. 1000, 500, 250, 125, ___, ___, ___

7. 2, 6, 12, 20, 30, ___, ___, ___

8. 1, 8, 27, 64, ___, ___, ___

B. Find the Rule

1. 5, 15, 45, 135, ... Rule:

2. 200, 185, 170, 155, ... Rule:

3. 1, 3, 7, 15, 31, ... Rule:

4. 2, 5, 10, 17, 26, ... Rule:

5. 1, 2, 4, 7, 11, 16, ... Rule:

6. 3, 4, 6, 9, 13, 18, ... Rule:

C. Find the Missing Number

1. 7, 14, ___, 28, 35

2. 3, 9, 27, ___, 243

3. 1, 3, 6, ___, 15, 21

4. 64, 32, ___, 8, 4

5. 1, 1, 2, 3, ___, 8, 13

6. 10, ___, 40, 80, 160

D. Create a Sequence

1. Start at 7, add 6 each time. Write the first 8 terms.

2. Start at 2, multiply by 3 each time. Write the first 6 terms.

3. Start at 500, subtract 45 each time. Write the first 7 terms.

4. Start at 1 and 1, use the Fibonacci rule. Write the first 12 terms.

5. Write the first 10 triangular numbers.

E. Square and Cube Numbers

1. Write the square numbers from 1² to 15².

2. Write the cube numbers from 1³ to 10³.

3. Which square numbers are also triangular numbers? (Hint: check the first 10 of each)

4. What is the difference between 12² and 11²? Between 20² and 19²? Do you see a pattern?

5. Is 144 a square number? Is 125 a cube number? Explain.

F. Multiple Choice Questions

1. The next term in 5, 10, 20, 40, ... is:

a) 50b) 60c) 80d) 100

2. The common difference in 3, 11, 19, 27, ... is:

a) 7b) 8c) 9d) 6

3. In the Fibonacci sequence, the number after 21 is:

a) 28b) 32c) 34d) 42

4. The 6th triangular number is:

a) 15b) 18c) 21d) 24

5. In a geometric pattern with first term 4 and common ratio 5, the 3rd term is:

a) 20b) 80c) 100d) 500

G. Word Problems

1. Meena saves ₹50 in the first week, ₹100 in the second, ₹150 in the third, and so on. How much will she save in the 10th week? What is her total savings after 10 weeks?

2. A ball bounces to half its height each time. If dropped from 128 cm, write the bounce heights as a sequence. After how many bounces will it be less than 5 cm?

3. A colony of bacteria doubles every hour. Starting with 5 bacteria, how many will there be after 8 hours?

4. In a competition, the 1st prize is ₹1,000, 2nd is ₹800, 3rd is ₹600, and so on. What is the prize for 7th place? How much total prize money is given to the top 7?

5. Arjun arranges chairs for a function: 8 in the 1st row, 11 in the 2nd, 14 in the 3rd, and so on. How many chairs are in the 10th row? How many chairs total in 10 rows?

H. Challenge: Magic Squares

1. Complete this magic square (magic constant = 15):

8	___	4
___	5	___
___	9	___

2. Make a 3×3 magic square using numbers 2, 4, 6, 8, 10, 12, 14, 16, 18 (magic constant = 30).

3. What is the pattern: 1² + 2² = ?, 1² + 2² + 3² = ?, 1² + 2² + 3² + 4² = ? Can you predict the sum of first 5 square numbers?

Activity: Pattern Detective

Instructions:

Find 5 patterns in your environment (floor tiles, fence posts, clock hours, calendar, Rangoli).
For each, write down the pattern as a number sequence.
Identify the type: arithmetic, geometric, repeating, or other.
Write the rule for each pattern.
Bonus: Create your own pattern and challenge a friend to find the rule!

Creative Challenge: Draw a Rangoli pattern that uses triangular numbers for the number of dots in each ring!

Image needed: An Indian child as a "Pattern Detective" with a magnifying glass, examining patterns: floor tiles, Rangoli dots, a clock face, and a calendar. Each has the number pattern written beside it.
Style: Fun detective theme, cartoon illustration, Indian context
Size: 650×400px

Want to review the lesson again?

Mathematics

Number Patterns and Sequences

Class 5 | CBSE / NCERT / ICSE

Name: Class: Date:

A pattern is a set of numbers that follow a definite rule. A sequence is an ordered list of numbers where each number follows that rule. Finding patterns helps us predict what comes next — a skill used in maths, science, music, and art!

Types of Number Patterns

Type	Rule	Example	Next 3 Terms
Arithmetic	Add or subtract a constant (common difference)	5, 10, 15, 20, ...	25, 30, 35
Arithmetic (−)	Subtract constant	100, 92, 84, 76, ...	68, 60, 52
Geometric (×)	Multiply by constant	2, 6, 18, 54, ...	162, 486, 1458
Geometric (÷)	Divide by constant	10000, 1000, 100, ...	10, 1, 0.1
Fibonacci	Next = sum of previous 2	1, 1, 2, 3, 5, 8, ...	13, 21, 34
Repeating	Group of numbers repeats	1, 4, 9, 1, 4, 9, ...	1, 4, 9

Special Number Sequences

Type	Sequence	Rule
Square Numbers	1, 4, 9, 16, 25, 36, 49, 64, 81, 100	n × n
Cube Numbers	1, 8, 27, 64, 125, 216, 343	n × n × n
Triangular Numbers	1, 3, 6, 10, 15, 21, 28, 36	Add 2, 3, 4, 5, ...
Fibonacci	0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...	Each = sum of prev 2

Triangular numbers shown as dot triangles: 1, 3, 6, 10, 15

How to Find the Rule

Step	What to Do	Example: 3, 7, 11, 15, ...
1	Find differences between terms	7−3=4, 11−7=4, 15−11=4
2	Constant difference → arithmetic	Yes! Common difference = +4
3	Not constant → find ratios	Not needed here
4	Constant ratio → geometric	—
5	Neither → two-step or special rule	—

Two-Step Rules

Example: 2, 5, 11, 23, 47, ... Rule: each term × 2 + 1

Check: 2×2+1=5 ✓, 5×2+1=11 ✓, 11×2+1=23 ✓. Next = 47×2+1 = 95

Magic Square (3×3, Magic Constant = 15)

2	7	6
9	5	1
4	3	8

Every row, column, and diagonal = 15. India's Khajuraho temple has a famous 4×4 magic square (constant = 34)!

Key Points: Arithmetic = add/subtract constant. Geometric = multiply/divide constant. Fibonacci: next = prev two added. Square numbers: differences are 3, 5, 7, 9, ... (odd numbers). Triangular: add 2, 3, 4, 5, ...

Fibonacci sequence in sunflower spirals and nature

Mathematics

Number Patterns — Practice

Class 5 | CBSE / NCERT / ICSE

A. Find the Next 3 Terms

1. 4, 9, 14, 19, ___, ___, ___

2. 100, 88, 76, 64, ___, ___, ___

3. 3, 6, 12, 24, ___, ___, ___

4. 1, 4, 9, 16, 25, ___, ___, ___

5. 1, 1, 2, 3, 5, 8, ___, ___, ___

6. 2, 6, 12, 20, 30, ___, ___, ___

7. 1, 8, 27, 64, ___, ___, ___

8. 1000, 500, 250, 125, ___, ___, ___

B. Find the Rule

1. 5, 15, 45, 135, ... Rule:

2. 200, 185, 170, 155, ... Rule:

3. 1, 3, 7, 15, 31, ... Rule:

4. 2, 5, 10, 17, 26, ... Rule:

5. 3, 4, 6, 9, 13, 18, ... Rule:

C. Find the Missing Number

1. 7, 14, ___, 28, 35

2. 3, 9, 27, ___, 243

3. 1, 3, 6, ___, 15, 21

4. 64, 32, ___, 8, 4

5. 1, 1, 2, 3, ___, 8, 13

6. 10, ___, 40, 80, 160

D. Multiple Choice

1. Next term in 5, 10, 20, 40, ...: a) 50b) 60c) 80d) 100

2. Common difference in 3, 11, 19, 27, ...: a) 7b) 8c) 9d) 6

3. After 21 in Fibonacci: a) 28b) 32c) 34d) 42

4. 6th triangular number: a) 15b) 18c) 21d) 24

Mathematics

Patterns — Word Problems & Activity

Class 5 | CBSE / NCERT / ICSE

E. Word Problems

1. Meena saves ₹50 in week 1, ₹100 in week 2, ₹150 in week 3, and so on. How much will she save in the 10th week? What is her total savings after 10 weeks?

2. A ball bounces to half its height each time. Dropped from 128 cm. Write the bounce heights as a sequence. After how many bounces will it be less than 5 cm?

3. Arjun arranges chairs: 8 in row 1, 11 in row 2, 14 in row 3, and so on. How many chairs in row 10? Total chairs in 10 rows?

4. Competition prizes: 1st = ₹1,000, 2nd = ₹800, 3rd = ₹600, and so on. Prize for 7th place? Total prize money for top 7?

5. A colony of bacteria doubles every hour, starting with 5. How many bacteria after 8 hours?

F. Challenge: Magic Squares

1. Complete this magic square (magic constant = 15):

8	___	4
___	5	___
___	9	___

2. What is the pattern: 1²=1, 1²+2²=5, 1²+2²+3²=14. Predict the sum of first 5 square numbers.

3. Create your own arithmetic pattern starting at 3, with common difference 7. Write the first 8 terms.

Activity: Pattern Detective

Find 5 patterns in your environment (floor tiles, Rangoli, clock, calendar, fence). Write each as a number sequence, identify the type (arithmetic/geometric/Fibonacci), and write the rule. Bonus: Draw a Rangoli using triangular numbers for dots in each ring!

Magic square from Khajuraho temple with magic constant 34

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