Class 6th ganit Prakash Chapter – 1
Class 6th ganit Prakash
Chapter – 1
Patterns in Mathematics
Q.1 What is Mathematics?
Ans. Mathematics aims not only to find out what patterns exist, but also to explain why they exist.
Patterns in Numbers
Example:
0, 1, 2, 3, 4, 5 …
• The branch of Mathematics that studies patterns in whole numbers is called Number Theory.
• Number sequences are the most basic and interesting types of patterns that mathematicians study.
Table-1: Examples of Number Sequences
1, 1, 1, 1, 1 … → All 1’s
1, 2, 3, 4, 5 … → Counting Numbers
1, 3, 5, 7, 9, 11 … → Odd Numbers
2, 4, 6, 8, 10 … → Even Numbers
1, 3, 6, 10, 15 … → Triangular Numbers
1, 4, 9, 16, 25 … → Square Numbers
1, 8, 27, 64 … → Cubes
1, 2, 3, 5, 8 … → Fibonacci Numbers
1, 2, 4, 8, 16 … → Powers of 2
1, 3, 9, 27 … → Powers of 3
Visualising Number Sequences
★ Pattern in Addition
1) Pattern by adding 3 consecutive numbers
1 + 2 + 3 = 6
2 + 3 + 4 = 9
3 + 4 + 5 = 12
4 + 5 + 6 = 15
…
Rule:
Sum of three consecutive numbers = 3 × middle number
2) Pattern by adding 4 consecutive numbers
1 + 2 + 3 + 4 = 10
2 + 3 + 4 + 5 = 14
3 + 4 + 5 + 6 = 18
4 + 5 + 6 + 7 = 22
…
Rule:
Sum of four consecutive numbers = 2 × (sum of the two middle numbers)
3) Pattern by adding odd numbers
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
Rule:
Sum of first n odd numbers = n² (square number)
**Pg – 3**
# Figure it Out
### Q1. Can you recognise the pattern in each sequence in Table–1?
**Ans:**
1. 1, 1, 1, 1, … → **All 1’s**
2. 1, 2, 3, 4, 5, … → **Counting Numbers**
3. 1, 3, 5, 7, 9, … → **Odd Numbers**
4. 2, 4, 6, 8, 10, … → **Even Numbers**
5. 1, 3, 6, 10, 15, … → **Triangular Numbers**
6. 1, 4, 9, 16, 25, … → **Square Numbers**
7. 1, 8, 27, 64, 125, … → **Cube Numbers**
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**Pg – 5**
# Figure it Out
### Q1. Copy the pictorial representation of sequences.
**Ans:** Same as **Table–2**.
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### Q2. Why are 1, 3, 6, 10, 15 called triangular numbers?
**Ans:**
The numbers **1, 3, 6, 10, 15** are called **triangular numbers** because they can be represented by a **triangular arrangement of dots**.
Example:
1
●
3
●
● ●
6
●
● ●
● ● ●
10
●
● ●
● ● ●
● ● ● ●
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### Q3. Why are 1, 4, 9, 16 called square numbers?
**Ans:**
The numbers **1, 4, 9, 16** are called **square numbers** because they can be represented by a **square arrangement of dots**.
Example:
1 → ●
4
● ●
● ●
9
● ● ●
● ● ●
● ● ●
16
●. ●. ● ●
● ● ● ●
● ● ● ●
● ● ●. ●
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Q.4 Why are 1, 8, 27, 64 called cube numbers?
Ans:
The numbers 1, 8, 27, 64 are called cube numbers because they can be arranged in the form of a cube using unit blocks.
Example:
1³ = 1 × 1 × 1 = 1
2³ = 2 × 2 × 2 = 8
3³ = 3 × 3 × 3 = 27
4³ = 4 × 4 × 4 = 64
Q.5 Pictorial way to visualise the sequence of power 2 ? Power of 3?
Answer
Sequence of power 2
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**Page
**Figure It Out**
**Q1. Can you think of other examples where mathematics helps us in our everyday life?**
**Ans.** Yes, mathematics helps us in many activities of our daily life. Some examples are:
1. **Shopping and budgeting** – We use mathematics to calculate prices, discounts, and manage money.
2. **Science and technology** – Mathematics is used in computers, machines, and scientific research.
3. **Navigation and travel** – It helps in finding directions, distances, and planning routes.
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**Q2. How has mathematics helped people understand the world better?**
**Ans.**
1. **Engineering and construction** – Mathematics helps engineers design safe buildings, bridges, and roads.
2. **Weather prediction** – It helps scientists predict weather and understand climate change.
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# Pg–2 Figure It Out
### Q5. What happens if we add pairs of consecutive triangular numbers?
**Solution:**
Add pairs of consecutive triangular numbers:
1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25
**Hence, we get a sequence of square numbers.**
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### Q6. Recognise the pattern in each sequence and find the next three numbers.
**(a)**
1, 4, 9, 16, 25, 36, 49, 64
Next three numbers: **81, 100, 121**
(Perfect square numbers)
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**(b)**
8, 27, 64, 125, 216
Next three numbers: **343, 512, 729**
(Cube numbers)
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**(c)**
1, 2, 3, 5, 8, 13, 21
Next three numbers: **34, 55, 89**
(Fibonacci pattern)
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**(d)**
1, 2, 4, 8, 16, 32, 64
Next three numbers: **128, 256, 512**
(Multiply by 2)
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**(e)**
1, 3, 9, 27, 81, 243, 729
Next three numbers: **2187, 6561, 19683**
(Multiply by 3)
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