Class 6 NCERT Solutions (2024-2025) Ganita Prakash Maths Chapter 1 Patterns in Mathematics
1.1 What is Mathematics?
Mathematics is the study of numbers, quantities, shapes, patterns, and logical relationships. It is both a science and a language that helps us describe, analyze, and solve problems in the real world.
Key Aspects of Mathematics:
Numbers & Calculations – Arithmetic (addition, subtraction, multiplication, division), algebra, and number theory.
Shapes & Spaces – Geometry, trigonometry, and measurements (area, volume, angles).
Patterns & Relationships – Algebra, functions, and sequences.
Data & Chance – Statistics (averages, graphs) and probability (predicting outcomes).
Logic & Reasoning – Problem-solving, proofs, and critical thinking.
Why is Mathematics Important?
It helps in daily tasks (shopping, cooking, time management).
It is essential in science, engineering, technology, and finance.
It improves logical thinking and problem-solving skills.
Mathematics in Nature & Real Life:
Symmetry in flowers and snowflakes.
Patterns in weather, music, and art.
Measurements in construction, medicine, and sports.
In short, mathematics is the foundation of understanding and organizing the world around us!
For example, the understanding of patterns in the motion of stars,
planets, and their satellites led humankind to develop the theory of
gravitation, allowing us to launch our own satellites and send rockets
to the Moon and to Mars; similarly, understanding patterns in genomes
has helped in diagnosing and curing diseases—among thousands of
other such examples
Figure it Out - Page number 2
1. Can you think of other examples where mathematics helps us in our everyday lives?
Solution:
Mathematics plays a crucial role in our everyday lives in many ways.Examples of Mathematics in Daily Life:
Budgeting & Shopping:
Calculating expenses, discounts, and managing savings.
Comparing prices to find the best deals.
Time Management:
Planning schedules, estimating travel time, and setting alarms.
Cooking & Baking:
Measuring ingredients, adjusting recipe quantities, and setting cooking times.
Travel & Navigation:
Calculating distances, fuel consumption, and reading maps/GPS.
Home Improvement:
Measuring rooms for furniture, paint, or flooring.
Calculating areas and volumes for construction.
Banking & Finance:
Calculating interest on loans/savings, managing investments, and understanding taxes.
Sports & Fitness:
Keeping score in games, tracking calories, and measuring workout progress.
Technology & Gadgets:
Using smartphones, computers, and apps that rely on mathematical algorithms.
Health & Medicine:
Measuring body temperature, blood pressure, and medication dosages.
Weather Forecasting:
Understanding temperature, humidity, and rainfall predictions.
Conclusion:
Mathematics plays a vital role in various professions, such as:
- Helping vegetable sellers calculate change accurately
- Enabling pot makers to craft pots of precise dimensions
- Assisting painters in estimating time and materials needed for a project
- Guiding masons in calculating the required number of bricks
- Aiding doctors in determining the correct dosage of medication
Mathematics is an essential tool in many real-world applications
Mathematics is everywhere—from simple daily tasks to complex decision-making. It helps us solve problems efficiently and make informed choices.
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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.).
Solution:
Mathematics has been the backbone of human progress, propelling advancements in science, technology, engineering, and society. Here are some key ways it has driven humanity forward:
1. Scientific Discovery & Experimentation
Physics & Astronomy: Mathematics enabled Newton’s laws of motion, Einstein’s theory of relativity, and space exploration (e.g., calculating rocket trajectories).
Medicine: Statistical models help in drug trials, epidemiology (e.g., predicting disease spread), and medical imaging (MRI, CT scans).
Chemistry: Equations predict reactions, helping design new materials, medicines, and energy solutions.
2. Engineering & Infrastructure
Bridges & Buildings: Calculus and geometry ensure stable structures (e.g., suspension bridges, skyscrapers).
Transportation: Algorithms optimize traffic flow; equations design safer cars, planes, and high-speed trains.
Electricity & Electronics: Maxwell’s equations underpin electrical engineering; binary math powers computers.
3. Technology & Communication
Computers & AI: Binary math, algorithms, and cryptography drive computing and machine learning.
Internet & Phones: Data compression (e.g., JPEG, MP3) and error-correcting codes enable fast, reliable communication.
GPS & Navigation: Relies on trigonometry and relativity to pinpoint locations accurately.
4. Economics & Governance
Finance & Markets: Probability and statistics guide investments, risk assessment, and economic policies.
Democracy & Voting Systems: Game theory helps design fair voting mechanisms and prevent manipulation.
Logistics & Supply Chains: Optimization math ensures efficient delivery of goods (e.g., Amazon, FedEx).
5. Everyday Life Innovations
Clocks & Calendars: Astronomy-based math keeps time accurate (leap years, time zones).
Consumer Tech: Math designs TVs (pixel algorithms), cameras (image processing), and even bicycles (gear ratios).
Weather Forecasting: Differential equations and supercomputers predict storms and climate trends.
Conclusion
Mathematics has helped propel humanity forward by Scientific Discovery, Technology, Economic and growth.
Mathematics is the "language of the universe," allowing us to model reality, solve problems, and innovate. Without it, modern civilization—from smartphones to space travel—would not exist.
1.2 Patterns in Numbers
Among the most basic patterns that occur in mathematics are
patterns of numbers, particularly patterns of whole numbers:
0, 1, 2, 3, 4, ...
The branch of Mathematics that studies patterns in whole
numbers is called number theory.
Number sequences are the most basic and among the most
fascinating types of patterns that mathematicians study.
Some key number sequences that are studied in Mathematics.
1. All 1’s (1, 1, 1, 1, ...)
Q: What is the next term in the sequence: 1, 1, 1, 1, 1, ...?
A: 1
2. Counting Numbers (1, 2, 3, 4, ...)
Q: What is the 10th term in the sequence of counting numbers?
A: 10
3. Odd Numbers (1, 3, 5, 7, ...)
Q: What is the 6th odd number?
A: 11
4. Even Numbers (2, 4, 6, 8, ...)
Q: What is the sum of the first 3 even numbers?
A: 2 + 4 + 6 = 12
5. Triangular Numbers (1, 3, 6, 10, ...)
Q: How do you generate the next triangular number after 10?
A: Add the next counting number (5) → 10 + 5 = 15
6. Square Numbers (1, 4, 9, 16, ...)
Q: What is the 7th square number?
A: 7² = 49
7. Cube Numbers (1, 8, 27, 64, ...)
Q: What is the cube of 5?
A: 5³ = 125
8. Fibonacci (Virahānka) Numbers (1, 2, 3, 5, 8, 13, ...)
Q: What is the next Fibonacci number after 13?
A: 8 + 13 = 21
9. Powers of 2 (1, 2, 4, 8, 16, ...)
Q: What is 2 raised to the power of 6?
A: 64
10. Powers of 3 (1, 3, 9, 27, ...)
Q: What is 3⁵ (3 to the 5th power)?
A: 243
Bonus Challenge Questions:
Q: Which sequence has terms that are sums of the previous two numbers?
A: Fibonacci (Virahānka) numbersQ: Which sequence represents numbers that can form equilateral triangles?
A: Triangular numbersQ: What is the difference between consecutive square numbers? (e.g., 4 - 1 = 3, 9 - 4 = 5, etc.)
A: The differences are consecutive odd numbers.
Figure it Out Page Number 3
1. Can you recognize the pattern in each of the sequences in Table 1?
Solution: Recognizing the Patterns in Each Sequence
| Sequence | Pattern (Rule) |
|---|---|
| 1, 1, 1, 1, 1, ... | All 1’s – Every term is 1. |
| 1, 2, 3, 4, 5, ... | Counting numbers – Start at 1, increase by 1 each time. |
| 1, 3, 5, 7, 9, ... | Odd numbers – Start at 1, increase by 2 each time. |
| 2, 4, 6, 8, 10, ... | Even numbers – Start at 2, increase by 2 each time. |
| 1, 3, 6, 10, 15, ... | Triangular numbers – Add the next counting number (e.g., 10 + 5 = 15). |
| 1, 4, 9, 16, 25, ... | Square numbers – Each term is n² (1², 2², 3², ...). |
| 1, 8, 27, 64, 125, ... | Cube numbers – Each term is n³ (1³, 2³, 3³, ...). |
| 1, 2, 3, 5, 8, 13, ... | Fibonacci (Virahānka) numbers – Each term is the sum of the two before it. |
| 1, 2, 4, 8, 16, 32, ... | Powers of 2 – Each term is 2ⁿ⁻¹ (2⁰=1, 2¹=2, 2²=4, ...). |
| 1, 3, 9, 27, 81, 243, ... | Powers of 3 – Each term is 3ⁿ⁻¹ (3⁰=1, 3¹=3, 3²=9, ...). |
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.
Solution: Extending Each Sequence (Next 3 Numbers) + Rule in My Own Words
(a) All 1’s ( 1, 1, 1, 1, 1, 1, 1 ...)
Next 3 terms: 1, 1, 1
Rule: Every number in the sequence is just 1—it never changes!
(b) Counting Numbers ( 1, 2, 3, 4, 5, 6, 7 ...)
Next 3 terms: 8, 9, 10
Rule: Start at 1, then keep adding 1 each time.
(c) Odd Numbers (1, 3, 5, 7, 9, 11 , 13 ...)
Next 3 terms: 15, 17, 19
Rule: Start at 1, then add 2 each time.
(d) Even Numbers ( 2,4, 6, 8, 10, 12, 14, .......)
Next 3 terms: 16, 18, 20
Rule: Start at 2, then add 2 each time.
(e) Triangular Numbers (1, 3, 6, 10, 15, 21, 28 ...)
Next 3 terms: 36, 45, 55
Rule: Start at 1, then add 2, then 3, then 4, and so on (each time adding the next counting number).
(f) Square Numbers ( 1, 4, 9, 16, 25,36,49,...)
Next 3 terms: 64, 81, 100
Rule: Each number is a perfect square (1×1, 2×2, 3×3, etc.).
(g) Cube Numbers (1, 8, 27, 64, 125, 216, 343, ...)
Next 3 terms: 512, 729,1000
Rule: Each number is a perfect cube (1×1×1, 2×2×2, 3×3×3, etc.).
(h) Fibonacci (Virahānka) Numbers (1, 2, 3, 5, 8, 13, 21,...)
Next 3 terms: 34, 55, 89
Rule: Start with 1 and 2, then each new number is the sum of the last two.
(i) Powers of 2 (1, 2, 4, 8, 16, 32, 64, ...)
Next 3 terms: 128, 256, 512
Rule: Each number is double the previous one (or 2 raised to increasing powers).
(j) Powers of 3 (1, 3, 9, 27, 81, 243, 729, ...)
Next 3 terms: 2187, 6561, 19683
Rule: Each number is triple the previous one (or 3 raised to increasing powers).
1.3 Visualising Number Sequences
Pictorial representation of some number sequences:
(Pictorial Number Sequences)
1. All 1’s (1, 1, 1, 1, ...)
Q: If this sequence continues forever, will any term ever be different from 1?
A: No, every term is always 1.
2. Counting Numbers (1, 2, 3, 4, ...)
Q: How would the 5th term in this sequence be represented pictorially?
A: A group of 5 objects (e.g., 5 dots or 5 sticks).
3. Odd Numbers (1, 3, 5, 7, ...)
Q: If the sequence represents "dots in stacked L-shapes," how many dots form the 4th L-shape?
A: 7 dots (since the 4th odd number is 7).
4. Even Numbers (2, 4, 6, 8, ...)
Q: If these numbers represent pairs of shoes, how many shoes are there in the 5th term?
A: 10 shoes (5th even number = 10).
5. Triangular Numbers (1, 3, 6, 10, ...)
Q: If each term forms a triangle with dots, how many dots are added to go from the 3rd to the 4th triangle?
A: 4 dots (3rd term = 6 dots, 4th term = 10 dots → 10 – 6 = 4).
6. Square Numbers (1, 4, 9, 16, ...)
Q: If each square number is a grid of dots (e.g., 1×1, 2×2), how many dots form the 5th square?
A: 25 dots (5 × 5 grid).
7. Cube Numbers (1, 8, 27, 64, ...)
Q: If cubes are represented as 3D stacks of blocks, how many blocks make the 3rd cube?
A: 27 blocks (3 × 3 × 3).
Bonus Challenge Questions
Q: Which sequence would form a perfect staircase when drawn as dots?
A: Triangular numbers (each layer adds one more dot than the last).Q: If "odd numbers" were represented as nested L-shapes, what would the 5th L-shape look like?
A: A larger L with 9 dots (5th odd number = 9).Q: How are square numbers visually different from triangular numbers?
A: Squares form equal rows and columns (e.g., 4×4), while triangles form staircase layers (1, 2, 3, ... dots per row).
Figure it Out Page number 5
1. Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!
| Sequence | Visual Pattern | Next Term Drawing |
|---|---|---|
| All 1’s | ● (single dot) | ● |
| Counting numbers | ● → ●● → ●●● → ●●●● → ●●●●● | ●●●●●● (6 dots) |
| Odd numbers | L-shapes: 1, 3, 5, 7 ,9 dots | ●●●●● ●●●●●● L with 11 dots (6th odd number) |
| Even numbers | Pairs: ●● → ●●●● → ●●●●●● → ●●●●●●●● | ●●●●●●●●●● (10 dots) |
| Triangular numbers | △ layers: 1, 3, 6, 10 dots | △ with 15 dots (add a row of 5) |
| Square numbers | 1×1, 2×2, 3×3, 4×4 grids | 5×5 grid (25 dots) |
| Cube numbers | 1³=1 block, 2³=8 blocks, 3³=27 blocks | 4×4×4 cube (64 blocks) |
Solution:
Triangular numbers: These numbers can be arranged in the shape of
a triangle by placing dots in increasing rows.
Square numbers: These numbers can be arranged in the shape of a square by placing dots in equal rows and columns.
Cubes: These numbers can be arranged in the shape of a cube by
placing dots in three-dimensional layers
Triangular numbers (1, 3, 6, 10...):
Form perfect squares:
Rule: (e.g., dots).
Cube numbers (1, 8, 27, 64...):
Form 3D cubes with blocks:1 block → 2×2×2 → 3×3×3 → ...
Rule: (e.g., blocks).
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!
3. 36 as Both Triangular and Square
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... (8 rows)Square: 6×6 grid = 36 dots.
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Try with 1 or 1225 (next such number)!
4. Hexagonal Numbers (1, 7, 19, 37...)These are called hexagonal numbers because they can be arranged in a hexagon shape.
Visual: Nested hexagons:Next term: 61 (add 24 dots in outer hexagon).
Rule: .
Here is one possible way of thinking about Powers of 2:
5. Visualizing Powers of 2 and 3
Powers of 2: You can visualize the powers of 2 as squares where each
subsequent square has twice the number of smaller squares as the
previous one.
Powers of 3: You can visualize the powers of 3 as cubes, where each subsequent cube has three times the number of smaller cubes as the previous one.
Powers of 2 (1, 2, 4, 8...):
Double each time:
- Powers of 3 (1, 3, 9, 27...):
● → △ of 3 → 3×3 grid → 3×3×3 cube → ...
Key Takeaways
Same number, different shapes: 36 can be a triangle and a square.
Patterns are everywhere: Hexagons, cubes, and grids reveal hidden math.
Try it! Draw your own sequences (e.g., pentagonal numbers).
Fun Challenge: Find the next number after 36 that’s both triangular and square! (Hint: It’s 1225.)
1. Square Grid Template
*(For square numbers, powers of 2, and 36-as-square)*
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How to use:
Circle a 6×6 grid to show 36 as a square number.
Circle 1×1, 2×2, 3×3, etc. for square numbers.
For powers of 2: Start with 1 dot, then circle 2, 4, 8, etc.
2. Triangular Grid Template
*(For triangular numbers and 36-as-triangle)*
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How to use:
Fill rows to form triangles:
1st row: 1 dot → total = 1
2nd row: 2 dots → total = 3
...
8th row: 8 dots → total = 36 (triangular!)
3. Hexagonal Grid Template
(For hexagonal numbers: 1, 7, 19, 37...)
● ● ●● ● ● ●
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How to use:
1 dot: Center dot.
7 dots: Center + 6 neighbors (first ring).
19 dots: Add 12 dots in the next ring (total = 7 + 12).
Next term (61): Add 24 dots in the outer ring (37 + 24).
4. Cube Template
(For cube numbers: 1, 8, 27...)
Front Layer Middle Layer Back Layer
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
How to use:
1³ = 1: One block in the center.
2³ = 8: 2×2×2 blocks (all layers).
3³ = 27: 3×3×3 blocks (fill all layers).
Activity Solutions
1. Drawing the Next Pictures
Triangular numbers: Add a row of 5 dots → 15 dots total.
Square numbers: Draw a 5×5 grid → 25 dots.
Cubes: Build a 4×4×4 cube → 64 blocks.
2. Why "Triangular," "Square," "Cubes"?
Triangular: Form perfect triangles (e.g., bowling pins).
Square: Form perfect squares (like a chessboard).
Cubes: Form 3D cubes (like Rubik’s cubes).
3. 36 as Triangle + Square
Triangle: 8 rows (1+2+3+...+8 = 36).
Square: 6×6 grid.
Triangular 36 Square 36
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● ● ● ● ● ● ●#### **4. Hexagonal Numbers**
- **Next number:** 61 (37 + 24).
- **Pattern:** Add multiples of 6 (6, 12, 18, 24...).
#### **5. Powers of 2/3 Visualization**
- **Powers of 2:** Binary splitting!
1 → ●
2 → ● ●
4 → ● ● ● ●
8 → ● ● ● ● ● ● ● ●
- **Powers of 3:** Ternary branching!
1 → ●
3 → ●─●─●
9 → 3 branches, each splitting into 3### **Bonus Challenge**
Find the next number after 36 that’s both triangular *and* square!
**Hint:** It’s 1225 (triangular = 49th term, square = 35²).
1.4 Relations among Number Sequences
Example: What happens when we start adding up odd numbers? 1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36 . . .
This is a really beautiful pattern!
Why does this happen? Do you think it will happen forever?
The answer is that the pattern does happen forever. But why? As mentioned earlier, the reason why the pattern happens is just as important and exciting as the pattern itself1 + 3 + 5 + 7 + 9 + 11 = 36.
Because such a picture can be made for a square of any size, this
explains why adding up odd numbers gives square numbers.
By drawing a similar picture, can you say
what is the sum of the
first 10 odd numbers?
Now by imagining a similar picture, or by drawing it partially, as
needed, can you say
what is the sum of the first 100 odd numbers?
Figure it Out - Page 8
Great! Let's break down the rich mathematical exploration from the image and the questions one by one. The image you uploaded shows a 3D cube with color-coded layers and a circular dot arrangement, helping visualize square numbers and layered additions.
1. Why does 1 + 2 + 3 + ... + 2 + 1 give square numbers?
The picture shows a cube made from layered L-shapes wrapping around a central yellow cube.
This represents:
-
1 (center)
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Then add 3 sides of 1 cube → total = 1 + 2 = 3
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Add 3 sides of 2 cubes → 1 + 2 + 3 + 2 + 1
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Continue this growing pattern:
The shape visually builds a square layer-by-layer around a center.
Pattern Rule:
The sequence 1 + 2 + 3 + ... + 3 + 2 + 1 = n²
Because each layer wraps around the previous one symmetrically.
2. What's the value of 1 + 2 + ... + 100 + 99 + ... + 2 + 1?
This is a symmetric sum up to 100 and back down:
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Total terms: 199 (1 through 100, then back down through 99 to 1)
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The sum:
Why?
Using the square pattern: the visual model represents this as a square of side 100.
3. Add the All 1’s sequence up and down: What do you get?
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All 1's Up: 1, 1+1 = 2, 1+1+1 = 3, etc. → This gives:
1, 2, 3, 4, ...= counting numbers. -
All 1's Up and Down:
1, 2, 3, ..., n, ..., 3, 2, 1→ This gives:
1 + 2 + 3 + ... + n + ... + 3 + 2 + 1 = n²
So:
Adding the All-1s sequence up gives the counting numbers.
Adding it up and down gives perfect squares.
4. Add Counting Numbers Up – What Do You Get?
The sum:
These are triangular numbers:
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1 → 1
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1 + 2 → 3
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1 + 2 + 3 → 6
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1 + 2 + 3 + 4 → 10
-
...
Pictorially: A triangle growing with rows:
Row 1: •
Row 2: ••
Row 3: •••
= triangle!
5. Add consecutive triangular numbers: 1+3, 3+6, 6+10...
Example:
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1 + 3 = 4
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3 + 6 = 9
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6 + 10 = 16
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10 + 15 = 25
→ Sequence: 4, 9, 16, 25... = perfect squares
Why?
Each pair of consecutive triangular numbers adds up to a square:
Visual: Two adjoining triangles fill a square.
6. Powers of 2: 1, 1+2, 1+2+4... then add 1
-
Powers of 2 sum:
1, 3, 7, 15, 31, 63, ... -
Add 1 to each:
2, 4, 8, 16, 32, 64, ...
So:
1 + 2 + 4 + ... + 2^n = 2^{n+1} - 1
Adding 1 gives 2^{n+1}
Explanation: This is the sum of a geometric series. Visually: binary tree or doubling boxes.
7. Multiply triangular numbers by 6 and add 1:
Triangular numbers: 1, 3, 6, 10, 15, 21, ...
Multiply by 6 and add 1:
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6×1 + 1 = 7
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6×3 + 1 = 19
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6×6 + 1 = 37
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6×10 + 1 = 61
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6×15 + 1 = 91
→ Sequence: 7, 19, 37, 61, 91...
**This is the sequence of centered hexagonal numbers!
Visual: A hexagon growing around a center point.
8. Add hexagonal numbers: 1, 1+7, 1+7+19, 1+7+19+37...
Hexagonal numbers:
1, 7, 19, 37, 61, 91, ...
Partial sums:
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1
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1 + 7 = 8
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1 + 7 + 19 = 27
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1 + 7 + 19 + 37 = 64
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...
Noticing:
→ 8 = 2² + 2²
→ 27 = 3³
→ 64 = 4³
→ These might approach cube numbers!
Explore further to confirm.
9. Find Your Own Patterns in Table 1
Here are a few ideas:
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Try differences between terms — are they increasing linearly (arithmetic) or multiplicatively (geometric)?
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Visualize sums in shapes: squares, triangles, pyramids.
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Connect powers (2ⁿ, n², n³) to areas or volumes.
You can find many sequences like:
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Fibonacci numbers
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Centered polygonal numbers
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Figurate numbers
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