Which diagram correctly shows the next step in the sequence of adding odd numbers to form a square?

A 4x4 square with 16 dots, formed by adding 7 dots to a 3x3 square.

What type of number sequence is formed by adding up pairs of consecutive triangular numbers (e.g., 1+3, 3+6, 6+10)?

Square numbers (1+3=4, 3+6=9, 6+10=16)

If the first term of a sequence is 1 and each subsequent term is obtained by adding the current term to the previous term, what is the 5th term?

This rule is unclear or incorrect.

Assertion (A): The sum of the first 'n' odd numbers is always a square number. Reason (R): A square number can be pictorially represented by arranging dots in a square grid.

A and R both correct, R explains A

Consider the sequence of powers of 2: 1, 2, 4, 8, 16, ... If you start adding these numbers sequentially (1, 1+2, 1+2+4, 1+2+4+8, ...), what sequence do you get? What happens if you then add 1 to each number in this new sequence?

1, 3, 7, 15, ...; then powers of 2

Counting numbers; then odd numbers

Triangular numbers; then square numbers

1, 3, 7, 15, ...; then powers of 2

If the number of dots in a pattern is 36, and it can be arranged perfectly both in a triangle and in a square, what does this imply about the number 36?

36 is both a triangular number and a square number.

Examine the given diagram showing the first few terms of a shape sequence. What would be the number of components (e.g., lines or dots) in the 4th term of this sequence, assuming it follows the pattern of 'Complete Graphs'?

10 (A complete graph with 5 vertices has 10 lines)

Is this CBSE Class 6 Maths Chapter 1 quiz free?

Yes, this quiz on 'Patterns in Mathematics' is completely free for all students. No registration or payment is required.

How many questions are in the quiz?

The main 'Game Mode' quiz contains 40 questions covering various difficulty levels: easy, medium, and hard.

Is this quiz based on the NCERT syllabus?

Absolutely! All questions are carefully designed to align with the CBSE NCERT Class 6 Mathematics syllabus for Chapter 1, 'Patterns in Mathematics'.

Can I reattempt the quiz multiple times?

Yes, you can attempt the quiz as many times as you like to practice and improve your understanding of mathematical patterns.

What topics does this quiz cover?

This quiz covers number sequences (like odd, even, square, triangular numbers), shape sequences (like regular polygons), and the relationships between them, as taught in Chapter 1.

Are there explanations for the answers?

Yes, detailed explanations are provided for every question to help you understand the correct reasoning and learn from your mistakes.

Home › CBSE › Class 6 › Maths › PATTERNS IN MATHEMATICS

CBSE · Class 6 · 🧮 Maths · Chapter 1

PATTERNS IN MATHEMATICS

Number sequencesShape sequencesTriangular numbersSquare numbersOdd and Even numbersPowers of numbers

Chapter 1, 'Patterns in Mathematics', introduces students to the fascinating world of mathematical patterns found in numbers and shapes. It covers key number sequences like odd, even, square, triangular, and Virahānka numbers, along with their pictorial representations. The chapter also explores shape sequences such as regular polygons and stacked figures, highlighting the surprising relationships between number and shape patterns. Understanding these patterns is fundamental to developing strong mathematical reasoning and problem-solving skills.

Patterns in Mathematics ko Samajhna

Mathematics mein patterns ki khoj aur unke explanations ko samajhna bahut important hai. Yeh patterns humare aas-paas har jagah hote hain – nature mein, gharon mein, schools mein, aur sun, moon, stars ke movement mein bhi.

Mathematics as Art and Science: Mathematicians mathematics ko art aur science dono maante hain kyunki isme patterns ko discover karna aur unhe explain karna creative aur logical dono hota hai.
Importance of Explanation: Sirf pattern dhundhna hi kaafi nahi hai, us pattern ke peeche ka reason samajhna bhi utna hi important hai. Yeh explanations future applications mein help karti hain.
Everyday Life mein Patterns:
Nature: Flower petals, honeycomb, spiral shells mein patterns.
Time: Din aur raat ka cycle, seasons ka change.
Technology: Computer algorithms, architectural designs.

Patterns humari life ka ek integral part hain, aur mathematics humein unhe decode karne mein help karta hai.

Example: Traffic lights ka sequence (Red, Yellow, Green) ek pattern hai. Iske peeche ka reason traffic flow ko manage karna hai.

❗Important

Mathematics sirf calculations nahi, balki patterns ko dhundhna aur unhe samajhna bhi hai.

Number Sequences ka Introduction

Number sequences mathematics mein sabse basic patterns mein se ek hain. Yeh numbers ka ek ordered list hota hai jo ek specific rule follow karta hai.

Number Theory: Mathematics ki woh branch jo whole numbers ke patterns ko study karti hai, use number theory kehte hain.
Common Number Sequences:
All 1's: 1, 1, 1, 1, ... (Har term 1 hai)
Counting Numbers (Natural Numbers): 1, 2, 3, 4, ... (Har agla term previous term se 1 zyada hai)
Odd Numbers: 1, 3, 5, 7, ... (Har agla term previous term se 2 zyada hai; 2 se divisible nahi hote)
Even Numbers: 2, 4, 6, 8, ... (Har agla term previous term se 2 zyada hai; 2 se divisible hote hain)
Triangular Numbers: 1, 3, 6, 10, 15, ... (Dots ko triangle shape mein arrange karne se bante hain; difference increase hota hai: +2, +3, +4, ...)
Squares (Square Numbers): 1, 4, 9, 16, 25, ... (Numbers jo kisi integer ka square hote hain: \(1^2, 2^2, 3^2, ...\))
Cubes (Cube Numbers): 1, 8, 27, 64, 125, ... (Numbers jo kisi integer ka cube hote hain: \(1^3, 2^3, 3^3, ...\))
Virahānka Numbers (Fibonacci Sequence): 1, 2, 3, 5, 8, 13, ... (Har term previous two terms ka sum hota hai: \(F_n = F_{n-1} + F_{n-2}\))
Powers of 2: 1, 2, 4, 8, 16, 32, ... (Har term previous term ka double hota hai: \(2^0, 2^1, 2^2, ...\))
Powers of 3: 1, 3, 9, 27, 81, ... (Har term previous term ka triple hota hai: \(3^0, 3^1, 3^2, ...\))

Rule for finding next terms: Har sequence ka apna ek unique rule hota hai. Is rule ko identify karke hum next terms predict kar sakte hain.

Example: Odd numbers 1, 3, 5, 7, ... mein next three terms honge 9, 11, 13 kyunki rule hai 'add 2 to the previous term'.

📖Definition

Number Theory: Mathematics ki woh branch jo whole numbers ke patterns ko study karti hai.

⭐Remember

Har number sequence ek specific rule follow karta hai. Is rule ko dhundhna hi pattern recognition hai.

Number Sequences ko Visualise karna

Number sequences ko pictures ya diagrams ke through dekhna unhe samajhne ka ek bahut accha tareeka hai. Isse concepts aur patterns aur clear ho jaate hain.

Odd Numbers: Odd numbers ko L-shaped layers mein arrange karke visualize kar sakte hain. [IMAGE: visualizing_odd_numbers_figoddnumberpartition]
1 dot
1 + 3 = 4 dots (2x2 square)
1 + 3 + 5 = 9 dots (3x3 square)
Isse pata chalta hai ki pehle 'n' odd numbers ka sum \(n^2\) hota hai.
Triangular Numbers: Dots ko triangle shape mein arrange karke.
1st: 1 dot
2nd: 3 dots (1+2)
3rd: 6 dots (1+2+3)
4th: 10 dots (1+2+3+4)
Rule: \(T_n = \frac{n(n+1)}{2}\)
Square Numbers: Dots ko square shape mein arrange karke.
1st: 1 dot (1x1)
2nd: 4 dots (2x2)
3rd: 9 dots (3x3)
Rule: \(S_n = n^2\)
Cube Numbers: Blocks ko cube shape mein arrange karke (3D visualization).
1st: 1 block (1x1x1)
2nd: 8 blocks (2x2x2)
3rd: 27 blocks (3x3x3)
Rule: \(C_n = n^3\)
Powers of 2: [IMAGE: visualizing_powers_of_2_figpowersof2visualization]
Ek square ko half-half divide karte jaana. Ya fir blocks ko double karte jaana.

Visualisation se complex relationships bhi easy ho jaati hain.

💡Tip

Exam mein agar kisi number sequence ka rule samajh na aaye, toh dots ya shapes banakar visualize karne ki koshish karo. Isse pattern jaldi mil jaata hai.

Number Sequences ke Beech Relationships

Number sequences ek doosre se surprising ways mein related ho sakte hain. In relationships ko samajhna mathematics ki beauty hai.

Odd Numbers aur Square Numbers:
1 = 1 (\(1^2\))
1 + 3 = 4 (\(2^2\))
1 + 3 + 5 = 9 (\(3^2\))
1 + 3 + 5 + 7 = 16 (\(4^2\))
Rule: Pehle 'n' odd numbers ka sum hamesha \(n^2\) hota hai. Iska pictorial explanation [IMAGE: visualizing_odd_numbers_figoddnumberpartition] mein dekha ja sakta hai, jahan square grid ko L-shaped odd number layers mein partition kiya gaya hai.
Counting Numbers aur Square Numbers:
1 + 2 + 1 = 4 (\(2^2\))
1 + 2 + 3 + 2 + 1 = 9 (\(3^2\))
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 (\(4^2\))
Rule: Counting numbers ko up and down add karne se square numbers milte hain. Example: \(1 + 2 + ... + n + ... + 2 + 1 = n^2\).
All 1's Sequence ka Sum:
1
1 + 1 = 2
1 + 1 + 1 = 3
Isse Counting Numbers sequence milta hai.
Counting Numbers ka Sum (Triangular Numbers):
1
1 + 2 = 3
1 + 2 + 3 = 6
Isse Triangular Numbers sequence milta hai.
Consecutive Triangular Numbers ka Sum:
T_1 + T_2 = 1 + 3 = 4 (\(2^2\))
T_2 + T_3 = 3 + 6 = 9 (\(3^2\))
T_3 + T_4 = 6 + 10 = 16 (\(4^2\))
Rule: Do consecutive triangular numbers ka sum hamesha ek Square Number hota hai.
Powers of 2 ka Sum:
1 = 2^1 - 1
1 + 2 = 3 = 2^2 - 1
1 + 2 + 4 = 7 = 2^3 - 1
1 + 2 + 4 + 8 = 15 = 2^4 - 1
Rule: Powers of 2 (starting from \(2^0\)) ka sum hamesha next power of 2 minus 1 hota hai. \(2^0 + 2^1 + ... + 2^{n-1} = 2^n - 1\).

In relationships ko samajhne se mathematical thinking develop hoti hai.

⭐Remember

Patterns sirf individual sequences mein nahi hote, balki sequences ke beech ke connections mein bhi hote hain.

❗Important

Adding first 'n' odd numbers always gives \(n^2\).

Geometric Shapes mein Patterns ko Explore karna

Numbers ki tarah, shapes mein bhi patterns hote hain. Mathematics ki woh branch jo shapes ke patterns ko study karti hai, use geometry kehte hain.

Dimensions of Shapes: Patterns 1D (line), 2D (flat shapes), 3D (solid shapes) ya usse bhi zyada dimensions mein ho sakte hain.
Shape Sequences ke Examples:
Regular Polygons: Triangle, Square, Pentagon, Hexagon, etc. (Sides aur angles equal hote hain).
3 sides (Triangle)
4 sides (Square/Quadrilateral)
5 sides (Pentagon)
6 sides (Hexagon)
... aur so on.
Complete Graphs: Points (vertices) ko connect karne wali lines (edges) ka pattern.
1 point: 0 lines
2 points: 1 line
3 points: 3 lines (Triangle)
4 points: 6 lines (Tetrahedron skeleton)
Stacked Squares: Squares ko ek specific pattern mein stack karna.
1 square
4 squares (2x2)
9 squares (3x3)
... yeh Square Numbers ka pattern dikhata hai.
Stacked Triangles: Triangles ko ek specific pattern mein stack karna.
1 triangle
3 triangles
6 triangles
... yeh Triangular Numbers ka pattern dikhata hai.
Koch Snowflake: Ek fractal shape jo har iteration mein aur complex hoti jaati hai. Har line segment ko ek 'speed bump' se replace kiya jaata hai.

Shapes mein patterns ko observe karna geometry ki foundation hai.

📖Definition

Geometry: Mathematics ki woh branch jo shapes ke patterns ko study karti hai.

⭐Remember

Regular polygons mein sides ki length aur interior angles equal hote hain.

Shape Sequences ko Number Sequences se Connect karna

Shape sequences aur number sequences ke beech bahut interesting relationships hote hain. Yeh relationships dono ko samajhne mein help karte hain.

Regular Polygons aur Counting Numbers:
Regular polygons ki number of sides ek counting number sequence banati hai (starting from 3): 3, 4, 5, 6, ... (Triangle, Square, Pentagon, Hexagon, ...).
Number of corners bhi same sequence hoti hai, kyunki har corner ke corresponding ek side hoti hai.
Complete Graphs aur Number of Lines:
Vertices (points) ki संख्या 'n' ho toh lines (edges) ki संख्या \(\frac{n(n-1)}{2}\) hoti hai.
1 point: 0 lines
2 points: 1 line
3 points: 3 lines
4 points: 6 lines
Yeh sequence Triangular Numbers se related hai (\(T_{n-1}\) for n points).
Stacked Squares aur Square Numbers:
Har stacked square arrangement mein total number of small squares ek square number sequence banati hai: 1, 4, 9, 16, ... (\(1^2, 2^2, 3^2, 4^2, ...\)).
Stacked Triangles aur Triangular Numbers:
Har stacked triangle arrangement mein total number of small triangles ek triangular number sequence banati hai: 1, 3, 6, 10, ....
Koch Snowflake aur Powers of 4:
Koch Snowflake ke har iteration mein number of line segments ka pattern hota hai: 3, 12, 48, ....
Yeh sequence \(3 \times 4^{n-1}\) ke form mein hai (3 times powers of 4).
Initial triangle mein 3 sides. First iteration mein har side 4 parts mein divide hoti hai, so \(3 \times 4 = 12\) sides. Second iteration mein \(12 \times 4 = 48\) sides.

Shapes aur numbers ka yeh connection mathematics ko aur bhi fascinating banata hai.

💡Tip

Jab bhi koi shape sequence dekho, uske elements (sides, corners, small units) ko count karke dekho. Aksar woh kisi known number sequence se match kar jaayega.

Easy (15)

Which branch of Mathematics primarily studies patterns in whole numbers?
- Geometry
- Algebra
- Number theory (correct)
- Calculus
Why: Number theory is the branch dedicated to studying patterns in whole numbers.
What are the next three numbers in the sequence: 1, 3, 5, 7, ...?
- 8, 9, 10
- 9, 11, 13 (correct)
- 7, 9, 11
- 10, 12, 14
Why: The sequence consists of consecutive odd numbers, so the next terms are 9, 11, 13.
Which of the following is a sequence of square numbers?
- 1, 2, 3, 4, ...
- 1, 3, 6, 10, ...
- 1, 4, 9, 16, ... (correct)
- 1, 8, 27, 64, ...
Why: Square numbers are obtained by multiplying a whole number by itself (1x1, 2x2, 3x3, etc.).
The sequence 1, 2, 3, 5, 8, 13, 21, ... is known as Virahānka numbers.
Why: This sequence, where each number is the sum of the two preceding ones, is indeed called Virahānka numbers.
Mathematics is considered both an art and a science.
Why: The text states mathematicians think of mathematics as both an art and a science due to its creative and explanatory nature.
The sequence 1, 3, 6, 10, 15, ... are called __________ numbers.
Why: These numbers can be visually arranged to form triangles.
The sequence 2, 4, 6, 8, 10, ... consists of ________ numbers.
Why: Numbers divisible by 2 are called even numbers.
What is the next number in the sequence: 1, 2, 4, 8, 16, ...?
- 20
- 24
- 32 (correct)
- 36
Why: This is a sequence of powers of 2, where each term is double the previous one.
The sum of the first two odd numbers (1 + 3) is a square number.
Why: 1 + 3 = 4, which is 2 squared (2²).
The term 'geometry' refers to the branch of Mathematics that studies patterns in _________.
Why: Geometry is the study of shapes, their properties, and spatial relationships.
The sequence 1, 8, 27, 64, ... consists of cube numbers.
Why: These numbers are obtained by multiplying a whole number by itself three times (1³, 2³, 3³, etc.).
What is the next number in the sequence: 1, 3, 9, 27, ...?
- 30
- 36
- 81 (correct)
- 54
Why: This sequence represents powers of 3, where each term is three times the previous one.
The term 'regular' in 'regular polygon' means that the shape has equal-length sides and equal angles.
Why: A regular polygon is defined by having all sides and all angles equal.
The sequence 1, 2, 3, 4, 5, ... is known as the __________ numbers.
Why: These are the natural numbers used for counting.
Which of these is NOT a type of number sequence mentioned in the chapter?
- Odd numbers
- Prime numbers (correct)
- Triangular numbers
- Powers of 2
Why: Prime numbers are not explicitly listed in Table 1 as a key number sequence studied in this chapter.

Medium (15)

If you add the first 'n' odd numbers, what kind of number do you get?
- Triangular number
- Even number
- Square number (correct)
- Cube number
Why: Adding the first 'n' odd numbers always results in n² (a square number).
Match the number sequence with its description.
Why: Each sequence has a specific name based on its pattern.
What is the rule for forming the numbers in the sequence: 1, 4, 9, 16, 25, ...?
Why: Each number is the square of its position in the sequence.
Arrange the following shapes in the order they appear in the sequence of Regular Polygons by increasing number of sides: Hexagon, Triangle, Square, Pentagon.
Why: Regular polygons are named by their number of sides: Triangle (3), Square (4), Pentagon (5), Hexagon (6).
Which of the following diagrams represents the 4th triangular number?
- A triangle with 6 dots
- A triangle with 10 dots (correct)
- A triangle with 15 dots
- A square with 16 dots
Why: The 4th triangular number is 10, formed by arranging 10 dots in a triangle.
What is the sum of the first 5 odd numbers (1 + 3 + 5 + 7 + 9)?
- 20
- 25 (correct)
- 30
- 36
Why: The sum of the first 'n' odd numbers is n². For n=5, the sum is 5² = 25.
Match the sequence with its name.
Why: These are common number sequences with specific names.
What is the next term in the sequence of Stacked Squares shown in Table 3?
Why: The number of squares in Stacked Squares follows the pattern of square numbers.
Arrange the following powers of 2 in ascending order: 16, 2, 64, 4, 32.
Why: Ascending order means from smallest to largest.
Which diagram correctly shows the next step in the sequence of adding odd numbers to form a square?
- A 3x3 square with 9 dots.
- A 4x4 square with 16 dots, formed by adding 7 dots to a 3x3 square. (correct)
- A 5x5 square with 25 dots.
- A 2x2 square with 4 dots.
Why: The pattern of adding odd numbers to form squares means adding 7 dots to a 3x3 square to get a 4x4 square.
What type of number sequence is formed by adding up pairs of consecutive triangular numbers (e.g., 1+3, 3+6, 6+10)?
- Odd numbers
- Even numbers
- Square numbers
- Square numbers (1+3=4, 3+6=9, 6+10=16) (correct)
Why: Adding consecutive triangular numbers (1+3=4, 3+6=9, 6+10=16) generates square numbers.
Match the sequence with its general term or property.
Why: Each sequence has a distinct pattern for generating its terms.
If the first term of a sequence is 1 and each subsequent term is obtained by adding the current term to the previous term, what is the 5th term?
- 5
- 8
- 13
- This rule is unclear or incorrect. (correct)
Why: The rule 'adding the current term to the previous term' is ambiguous for generating a sequence like Virahānka numbers.
Which of the following diagrams correctly represents the 3rd square number?
- A 2x2 square with 4 dots.
- A 3x3 square with 9 dots. (correct)
- A 4x4 square with 16 dots.
- A triangle with 6 dots.
Why: The 3rd square number is 3², which is 9, represented by a 3x3 array of dots.
What is the number of sides in a regular pentagon?
- 4
- 5 (correct)
- 6
- 7
Why: A pentagon is a polygon with five sides.

Hard (10)

What is the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1?
Why: This sum is equivalent to 100², which is 10,000, following the pattern of square numbers.
Assertion (A): The sum of the first 'n' odd numbers is always a square number. Reason (R): A square number can be pictorially represented by arranging dots in a square grid.
Why: Both statements are true, and the pictorial representation of square numbers explains why adding odd numbers forms squares.
Consider the sequence of powers of 2: 1, 2, 4, 8, 16, ... If you start adding these numbers sequentially (1, 1+2, 1+2+4, 1+2+4+8, ...), what sequence do you get? What happens if you then add 1 to each number in this new sequence?
- Counting numbers; then odd numbers
- Triangular numbers; then square numbers
- 1, 3, 7, 15, ...; then powers of 2 (correct)
- Even numbers; then powers of 3
Why: The sums are 1, 3, 7, 15, ... Adding 1 to these gives 2, 4, 8, 16, ... which are powers of 2.
If the number of dots in a pattern is 36, and it can be arranged perfectly both in a triangle and in a square, what does this imply about the number 36?
- 36 is a prime number.
- 36 is both a triangular number and a square number. (correct)
- 36 is an odd number.
- 36 is a cube number.
Why: The chapter highlights 36 as a unique number that is both triangular and square.
The Koch Snowflake sequence starts with 3 line segments. Each segment is replaced by 4 smaller segments in the next iteration. If the sequence of total line segments is 3, 12, 48, ..., what will be the total number of line segments in the 5th shape of the sequence?
Why: The sequence is 3, 12, 48, 192, 768. Each term is multiplied by 4.
Assertion (A): Visualizing mathematical objects through pictures can be a very fruitful way to understand mathematical patterns. Reason (R): Pictures always provide a complete and rigorous proof for mathematical patterns.
Why: While visualization is helpful for understanding, pictures alone do not always provide rigorous mathematical proofs.
The sequence of hexagonal numbers starts with 1, 7, 19, 37, ... What is the next number in this sequence?
Why: The differences are 6, 12, 18, so the next difference is 24. Thus, 37 + 24 = 61.
Consider the sequence of 'Stacked Triangles' from Table 3. If the first shape has 1 triangle, the second has 3, and the third has 6, how many little triangles would be in the 5th shape of this sequence?
Why: The sequence of Stacked Triangles corresponds to triangular numbers. The 5th triangular number is 15.
Examine the given diagram showing the first few terms of a shape sequence. What would be the number of components (e.g., lines or dots) in the 4th term of this sequence, assuming it follows the pattern of 'Complete Graphs'?
- 6
- 10
- 15
- 10 (A complete graph with 5 vertices has 10 lines) (correct)
Why: The number of lines in a complete graph with 'n' vertices is a triangular number. For 5 vertices (4th term), it's the 4th triangular number, which is 10.
Observe the pattern of dots forming a sequence. If the first term has 1 dot, the second has 4 dots, and the third has 9 dots, what is the total number of dots in the 6th term of this sequence?
- 25
- 30
- 36 (correct)
- 49
Why: The sequence 1, 4, 9, ... represents square numbers. The 6th square number is 6² = 36.