NCERT Solutions for Class 6 Maths Chapter 1 Patterns in Mathematics

Class 6 Maths Chapter 1, "Patterns in Mathematics," introduces students to the fun and exciting world of patterns. From numbers to shapes, this chapter helps young minds understand how certain things follow a fixed rule or sequence. These patterns are not only found in maths but also in real life—like in nature, designs, and everyday objects.

By learning patterns, students build a strong base for logical thinking and problem-solving, and with the help of NCERT Solutions for Class 6 Maths Chapter 1, it becomes easier for them to understand the rules behind different patterns and solve questions confidently.

Read More: Whole Numbers

Class 6 Maths Chapter 1 Overview

Class 6 Maths Chapter 1 helps students understand how to find and continue patterns in numbers and shapes. It teaches them to identify various types of patterns like even numbers, square numbers, and repeating shapes. Students also learn to draw and extend these patterns by following simple rules.

Maths class 6 chapter 1 also shows how patterns are seen in real life, like in nature, buildings, and art. It further explains how shadows change with light, making learning more engaging.

NCERT Class 6 Chapter 1 Question Answer

The NCERT Class 6 Chapter 1 question answer section includes a total of six exercises based on the topic "Patterns in Mathematics."

• In Exercise 1.1, students learn what mathematics is and how it helps in daily life.

• Exercise 1.2 teaches how to find and understand number patterns.

• Exercise 1.3 focuses on visualizing number sequences to see how they grow.

• In Exercise 1.4, students find connections between different number patterns.

• Exercise 1.5 helps students spot and create patterns using shapes.

• Lastly, Exercise 1.6 shows how number patterns and shape patterns are linked.

The NCERT class 6 maths solutions provided below are going to make it easier for students to understand each topic clearly and practice pattern-based problems confidently while doing homework or solving it in school.

Also Read: What is Slope Formula?

Exercise Wise Class 6 Maths Chapter 1 Solutions 

The Class 6 Maths Chapter 1 solutions provided here give clear and easy-to-understand answers for all exercises in the chapter. These solutions will help students learn each topic step-by-step and solve questions confidently.

Exercise 1.1

1. Can you think of other examples where mathematics helps us in our everyday lives?

Ans. Mathematics helps us in many daily activities, such as:

• Managing Money: Making budgets and keeping track of spending.

• Cooking and Baking: Measuring ingredients and adjusting recipe amounts.

• Smart Shopping: Calculating discounts and adding up prices.

• Home Projects: Measuring rooms and materials needed for repairs.

• Travel Planning: Finding distances and estimating how long trips will take.

• Health Tracking: Counting calories and monitoring exercise routines.

• Organizing Time: Planning tasks and daily schedules.

• Gardening Help: Spacing plants and designing garden areas.

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2. How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses, or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)

Ans. Mathematics has helped humans progress in many important ways:

• Scientific Experiments: Maths helps plan and understand experiments, like studying forces in physics.

• Economy and Democracy: Maths manages money systems and resources and helps analyze election results.

• Engineering and Building: Maths is used to design and make bridges, buildings, and other safe structures.

• Technology: Maths is needed to create TVs, phones, computers, and digital systems.

• Transportation: Maths helps design and run bikes, cars, trains, and planes by calculating speed and fuel use.

• Timekeeping: Maths is used to make accurate clocks and calendars.

• Medicine: Maths helps with medical scans and calculating the right medicine doses.

• Space Exploration: Maths calculates planet paths and helps send satellites and space missions.

Exercise 1.2

Table 1: Examples of number sequences.

Figure it Out

1. Can you recognise the pattern in each of the sequences in Table 1?

Ans. (a) 1, 1, 1, 1, 1, 1, 1, …
The number 1 is repeated continuously without any change.

(b) 1, 2, 3, 4, 5, 6, 7, …
These are counting numbers starting from 1. Each number increases by 1.

(c) 1, 3, 5, 7, 9, 11, 13, …
These are odd numbers. Starting from 1, 2 is added each time.

(d) 2, 4, 6, 8, 10, 12, 14, …
These are even numbers. Starting from 2, 2 is added each time.

(e) 1, 3, 6, 10, 15, 21, 28, …
These are triangular numbers. Each term is the sum of natural numbers up to that position.

(f) 1, 4, 9, 16, 25, 36, 49, …
These are square numbers. Each term is the square of its position (1², 2², 3², and so on).

(g) 1, 8, 27, 64, 125, 216, …
These are cube numbers. Each term is the cube of its position (1³, 2³, 3³, etc.).

(h) 1, 2, 3, 5, 8, 13, 21, …
These are Virahānka numbers (also called the Fibonacci sequence). Each term is the sum of the two previous terms.

(i) 1, 2, 4, 8, 16, 32, 64, …
These are powers of 2. Each term is 2 raised to the power of its position (2¹, 2², 2³, etc.).

(j) 1, 3, 9, 27, 81, 243, 729, …
These are powers of 3. Each term is 3 raised to the power of its position (3¹, 3², 3³, and so on).

2. Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.

Ans. (a) 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
This sequence repeats the number 1 again and again. The next three numbers are also 1.

(b) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …
These are counting numbers going up by 1 each time. The next three numbers are 8, 9, and 10.

(c) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …
This is a sequence of odd numbers increasing by 2. The next three numbers are 15, 17, and 19.

(d) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
This sequence has even numbers increasing by 2. The next three numbers are 16, 18, and 20.

(e) 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …
These are triangular numbers. Each number is the sum of natural numbers up to that point. 

• 28 + 8 = 36

• 36 + 9 = 45

• 45 + 10 = 55.

(f) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
These are square numbers formed by multiplying a number by itself. The next three numbers are 64 (8²), 81 (9²), and 100 (10²).

(g) 1, 8, 27, 64, 125, 216, 343, 512, 729, …
This is a sequence of cube numbers. Each number is the cube of a natural number. The next three are 343 (7³), 512 (8³), and 729 (9³).

(h) 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
This is the Fibonacci (Virahanka) sequence. Each number is the sum of the previous two. The next three numbers are 34, 55, and 89.

• 13 + 21 = 34

• 21 + 34 = 55

• 34 + 55 = 89

(i) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, …
These are powers of 2, where each number is multiplied by 2. The next three numbers are 128, 256, and 512.

• 64 × 2 = 128

• 128 × 2 = 256

• 256 × 2 = 512

(j) 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, …
These are powers of 3, where each number is multiplied by 3. The next three numbers are 2187, 6561, and 19683.

• 729 × 3 = 2187

• 2187 × 3 = 6561

• 6561 × 3 = 19683

Exercise 1.3

Let us represent the first seven sequences in Table 1 using the following pictures. 

Table 2: Pictorial representation of some number sequences.

Figure it Out:

1. Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!

2. Why are 1, 3, 6, 10, 15, … called triangular numbers? 

Why are 1, 4, 9, 16, 25, … called square numbers or squares? 

Why are 1, 8, 27, 64, 125, … called cubes?

Ans. 1, 3, 6, 10, 15, … called triangular numbers:

These numbers are called triangular numbers because you can arrange dots to form a triangle shape. For example, the number 3 can be shown as a triangle with 2 dots at the bottom and 1 dot on top. Each number in this sequence shows how many dots you need to make a triangle with more rows.

1, 4, 9, 16, 25, … called square numbers:

These are called square numbers because they show the area of a square. For example, 4 is the area of a square with each side measuring 2 units (2 × 2 = 4). Every number in this sequence is made by multiplying a number by itself, which is the same as finding the area of a square.

1, 8, 27, 64, 125, … called cubes:

These numbers are called cubes because they show the volume of a cube. For example, 8 is the volume of a cube with each side of length 2 units (2 × 2 × 2 = 8). Each number in the sequence is found by multiplying a number by itself two more times, giving the total space inside the cube.

3. You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways! 

Ans. Some numbers can be both triangular and square. Three examples are 1, 1225, and 41616. For example, 1225 is the 49th triangular number. This means you can arrange 1225 dots in a triangle with 49 rows.

At the same time, 1225 can also make a perfect square with 35 dots on each side because 35 × 35 = 1225. This shows that some numbers can be shown in different shapes, like both a triangle and a square. It is interesting how one number, like 1225, fits exactly into both these shapes, showing a special link between different math patterns.

4. What would you call the following sequence of numbers? 

That’s right, they are called hexagonal numbers! 

Draw these in your notebook. What is the next number in the sequence?

Ans. Let's look at the pattern in this sequence:

• The 1st number is 1.

• The 2nd number is 1 + 6 = 7 (we add 6 × 1).

• The 3rd number is 7 + 12 = 19 (we add 6 × 2).

• The 4th number is 19 + 18 = 37 (we add 6 × 3).

• The 5th number is 37 + 24 = 61 (we add 6 × 4).

So, each time, we add a number that is 6 times the position number (1, 2, 3, 4, ...).

To find the next number, add 6 × 5 = 30 to 61.

Next number = 61 + 30 = 91.

This shows the sequence increases by adding larger multiples of 6 each time.

3. Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?

Here is one possible way of thinking about Powers of 2:

Ans: Pictorial Representation for powers of 3:

Exercise 1.4

1. Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, …, gives square numbers? 

Ans. 

• The first term is 1 (which is 1²).

• The second term is 1 + 2 + 1 = 4 (which is 2²).

• The third term is 1 + 2 + 3 + 2 + 1 = 9 (which is 3²).

• The fourth term is 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 (which is 4²).

• The fifth term is 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 (which is 5²).

• The sixth term is 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 (which is 6²).

2. By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1? 

Ans.

1. Understanding the Pattern: The pattern begins by counting up from 1 to 100, and then it goes back down from 99 to 1. This creates a mirror-like or symmetrical pattern with 100 at the center.

2. The sum of Increasing Sequence:
   Sum₁ to ₁₀₀ = (100 × (100 + 1)) ÷ 2
   = (100 × 101) ÷ 2
   = 5050

3. The sum of Decreasing Sequence:
   Sum₁ to ₉₉ = (99 × (99 + 1)) ÷ 2
   = (99 × 100) ÷ 2
   = 4950

4. Add the Two Sums:
   Total Sum = 5050 + 100 + 4950 = 10000

So, the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1 is 10,000.

3. Which sequence do you get when you start to add the All 1’s sequence up? What sequence do you get when you add the All 1’s sequence up and down?

Ans. When you add a sequence of 1’s going up, like 1 + 1 + 1 + 1, you get 4. If you add the same 1’s going down, like 1 + 1 + 1 + 1 again, the total is still 4. So, whether you add the 1s from top to bottom or bottom to top, the answer stays the same.

4. Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?

Ans. 

When you keep adding the counting numbers one after another, you get a new sequence like this:

• Start with 1

• Then 1 + 2 = 3

• Then 1 + 2 + 3 = 6

• Then 1 + 2 + 3 + 4 = 10

So, the numbers you get are: 1, 3, 6, 10 and so on.

5. What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, …? Which sequence do you get? Why? Can you explain it with a picture?

Ans. When we add pairs of consecutive triangular numbers, we get a new sequence that matches the pentagonal numbers. Here's how it works:

Step 1: Know the triangular numbers. Triangular numbers are: 1, 3, 6, 10, 15, ...

Step 2: Add pairs of consecutive triangular numbers

• 1 + 3 = 4 → this is the 1st pentagonal number

• 3 + 6 = 9 → this is the 2nd pentagonal number

• 6 + 10 = 16 → this is the 3rd pentagonal number

• 10 + 15 = 25 → this is the 4th pentagonal number

Each time we add two triangular numbers one after the other, the total number of dots forms a shape like a pentagon. That’s why the result is called a pentagonal number. This pattern shows how dots can be arranged neatly to make a pentagon, just like triangular numbers form triangles.

6. What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, …? Now add 1 to each of these numbers—what numbers do you get? Why does this happen?

Ans. When we start adding powers of 2, we get a special pattern. Let’s understand it step by step.

Adding Powers of 2:

• First number: 1

• Next: 1 + 2 = 3

• Then: 1 + 2 + 4 = 7

• Next: 1 + 2 + 4 + 8 = 15

• Then: 1 + 2 + 4 + 8 + 16 = 31

So, the sequence we get is: 1, 3, 7, 15, 31…

These numbers are always 1 less than the next power of 2. For example, 3 is 1 less than 4, 7 is 1 less than 8, and so on. 

Now, add 1 to each of them:

• 1 + 1 = 2

• 3 + 1 = 4

• 7 + 1 = 8

• 15 + 1 = 16

• 31 + 1 = 32

So, we get: 2, 4, 8, 16, 32…, which is the sequence of powers of 2.

Explanation:

When we add powers of 2 one after another, the total is always just one less than the next power of 2. Here's why:

• The first few powers of 2 are written like this:
  2⁰ + 2¹ + 2² + ... + 2ⁿ⁻¹

• There is a formula to quickly find the total of this kind of sum. The formula is:
  Sum = 2ⁿ – 1

• If we add 1 to this total, we get:
  2ⁿ – 1 + 1 = 2ⁿ

This means, when we add 1 to the total, we get a power of 2. So, if we keep adding the powers of 2 and then add 1 to each sum, we get the sequence:
2, 4, 8, 16, 32, … 

This is the list of powers of 2.

7. What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture? 

Ans. Triangular numbers follow this pattern: 1, 3, 6, 10, 15, 21, and so on. Now, if we multiply each triangular number by 6 and then add 1, we get a new sequence:

• 1 × 6 + 1 = 7

• 3 × 6 + 1 = 19 (difference of 12)

• 6 × 6 + 1 = 37 (difference of 18)

• 10 × 6 + 1 = 61 (difference of 24)

• 15 × 6 + 1 = 91 (difference of 30)

So, the new sequence becomes: 7, 19, 37, 61, 91, ...

In this pattern, each new number increases by 6 more than the last difference. That means the gap between the numbers keeps growing by 6 each time.

8. What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, … ? Which sequence do you get? Can you explain it using a picture of a cube?

Ans. Hexagonal numbers follow this sequence: 1, 7, 19, 37, and so on. When you add these numbers one by one, you get special results:

• The first hexagonal number is 1, and its sum is 1, which is the cube of 1 (1³).

• When you add the second number 7, the total becomes 1 + 7 = 8, which is the cube of 2 (2³).

• Adding the third number, 19, the sum is 1 + 7 + 19 = 27, which is the cube of 3 (3³).

• Adding the fourth number, 37, the total is 1 + 7 + 19 + 37 = 64, which is the cube of 4 (4³).

• Adding the fifth number, 61, the sum becomes 1 + 7 + 19 + 37 + 61 = 125, which is the cube of 5 (5³).

This shows a cool pattern: the sum of the first n hexagonal numbers is always equal to n³, the cube of that number. It is a nice way to see how hexagonal numbers connect with perfect cubes.

9. Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?

Ans. Here are two easy patterns in the sequences:

1. Multiples of 3: The numbers 3, 6, 9, 12, 15, 18, and so on are all multiples of 3. Each number is 3 more than the one before it.

2. Starting at 10 and adding 5 each time: The sequence 10, 15, 20, 25, … begins at 10, and every next number goes up by 5.

In the first sequence, every term is just 3 times a whole number (like 1, 2, 3, 4…). In the second sequence, you start at 10 and keep adding 5 again and again. These show how simple rules can create clear and regular patterns.

If you draw these on a number line or graph, you will see the equal steps between numbers, which explains why these patterns happen.

Exercise 1.5

Table 3: Examples of shape sequences.

1. Can you recognise the pattern in each of the sequences in Table 3? 

Ans.(a) Regular Polygons: These are shapes like triangles, quadrilaterals, pentagons, and hexagons. In this pattern, the number of sides goes up by 1 each time, starting from 3. So, each polygon has one more side than the one before it, creating a simple number sequence.

(b) Complete Graphs

The number of lines in the sequence is: For K2 = 1, K3 = 3, K4 = 6, K5 = 10, and K6 = 15.

This gives the series 1, 3, 6, 10, 15, and so on. This is called a triangular number sequence because each number shows the total lines that can make a triangle.

Triangular numbers are made by adding natural numbers one after another, so the sequence grows in a simple and predictable way.

(c) Stacked Squares: The number of small squares in each layer follows this pattern: 1, 4, 9, 16, 25, and so on. These are called square numbers because each term is a natural number raised to the power of 2 (like 1², 2², 3², and so on).

The arrangement creates a perfect square shape when seen visually. This shows how the number of small squares grows as the layers increase, fitting neatly into a square grid. That’s why this pattern clearly represents a sequence of square numbers.

(d) Stacked Triangles: The number of small triangles in each layer follows the pattern:

So, this pattern is also a square number sequence, but shown using triangles instead of squares.

(e) Koch Snowflake: The number of sides in each stage becomes 4 times the number of sides in the previous stage.

The number of sides in each stage increases by a factor of 4.

2. Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.

Ans: (a) Regular Polygon

A polygon with 11 sides is known as a hendecagon.

(b) K6

The image shows a complete graph with 7 points (called K7), where each point is connected by a straight line to every other point.

(c) Stacked Squares

The total number of squares is 6 × 6 = 36. This shows a perfect square because the small squares make a grid with 6 rows and 6 columns.

(d) Stacked Triangles

The total number of triangles is 1 + 3 + 5 + 7 + 9 + 11 = 36. This shows that adding odd numbers one after another gives the total count of triangles in the arrangement.

(e) Koch Snowflake

The shape is a Koch snowflake, made by adding small triangular bumps to every side of an equilateral triangle again and again.

Exercise 1.6

1. Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens? 

Ans.

Both sequences match because in a regular polygon, the number of sides is exactly the same as the number of corners (vertices).

2. Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?

Ans.

So, the sequence goes 1, 3, 6, 10, 15, and continues like this. This type of sequence is called a triangular number sequence.

3. How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why? 

We get the sequence 1, 4, 9, 16, 25, 36, and so on. This sequence shows square numbers.

4. How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?) 

Ans.

The numbers 1, 4, 9, 16, 25, 36, and 49 are square numbers. If we add a stacked triangle shape at the bottom, it helps us find the next number in this square number sequence.

5. To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment ‘—’ by a ‘speed bump’ __ⴷ__. As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, ..., i.e., 3 times Powers of 4; this sequence is not shown in Table 1.)

The sequence 3, 12, 48, 192, 768, and so on starts with 3. To get each next number, you multiply the one before it by 4. In the same way, when the Koch Snowflake grows, it adds four new lines for every line that was in the shape before.

Practicing these questions helps you know important math patterns clearly. Use these NCERT solutions for Class 6 Maths Chapter 1 to check your answers and improve your understanding.

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Advantages of NCERT Solutions for Class 6 Maths Chapter 1 - Patterns in Mathematics

NCERT Solutions for Class 6 Maths Chapter 1 make it easier for students to learn about number patterns. They explain things simply and show how numbers follow certain rules to form clear patterns. Here are some benefits of using these solutions:

• They give easy and clear explanations about number patterns and sequences, helping students understand how numbers grow or change in a regular way.

• Step-by-step methods are provided to find and continue patterns, so students can learn and use these ideas more easily.

• These solutions build a strong base in number patterns, which is important for learning harder math topics later on.

• Many practice questions are included, which help students get better at identifying patterns and solving problems.

Read More: Real Numbers

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