Patterns in Whole Numbers
Have you noticed how tiles on a floor form patterns? Or how the seats in a cinema are arranged in rows and columns? Numbers also form beautiful patterns. When you arrange dots or objects in different shapes, some very interesting number patterns appear.
In this topic, you will discover number patterns that come from arranging objects in lines, triangles, squares, and rectangles. These patterns help you predict numbers, find shortcuts in calculations, and see the beauty hidden inside mathematics.
This is part of the Whole Numbers chapter in Class 6 NCERT Maths. Spotting patterns is one of the most useful skills in maths — once you see a pattern, you can often solve problems without doing long calculations.
What is Patterns in Whole Numbers - Grade 6 Maths (Whole Numbers)?
Definition: A number pattern is a sequence of numbers that follows a particular rule or arrangement.
Types of number patterns based on shapes:
- Line numbers: Numbers of dots that can be arranged in a straight line — all whole numbers (1, 2, 3, 4, 5, ...)
- Triangular numbers: Numbers of dots that can be arranged in the shape of a triangle — 1, 3, 6, 10, 15, 21, ...
- Square numbers: Numbers of dots that can be arranged in the shape of a square — 1, 4, 9, 16, 25, 36, ...
- Rectangular numbers: Numbers of dots that can be arranged in the shape of a rectangle (with more than 1 row and more than 1 column) — 6, 8, 10, 12, ...
Key idea: Some numbers fit more than one pattern. For example, 36 is both a square number (6 × 6) and a triangular number (it is the 8th triangular number).
Patterns in Whole Numbers Formula
Formulas for number patterns:
Triangular Numbers:
nth triangular number = n × (n + 1) / 2
Where n is the position (1st, 2nd, 3rd, ...).
- 1st triangular number = 1 × 2 / 2 = 1
- 2nd triangular number = 2 × 3 / 2 = 3
- 3rd triangular number = 3 × 4 / 2 = 6
- 4th = 10, 5th = 15, 6th = 21, 7th = 28, ...
Square Numbers:
nth square number = n × n = n²
- 1st square number = 1² = 1
- 2nd square number = 2² = 4
- 3rd square number = 3² = 9
- 4th = 16, 5th = 25, 6th = 36, ...
Amazing pattern in square numbers:
- 1 = 1
- 4 = 1 + 3
- 9 = 1 + 3 + 5
- 16 = 1 + 3 + 5 + 7
- 25 = 1 + 3 + 5 + 7 + 9
- Every square number is the sum of consecutive odd numbers starting from 1.
Derivation and Proof
How triangular numbers are built:
- Start with 1 dot → 1st triangular number = 1
- Add a row of 2 dots below → 1 + 2 = 3 (2nd triangular number)
- Add a row of 3 dots below → 1 + 2 + 3 = 6 (3rd triangular number)
- Add a row of 4 dots → 1 + 2 + 3 + 4 = 10 (4th triangular number)
- Add a row of 5 dots → 1 + 2 + 3 + 4 + 5 = 15 (5th triangular number)
Each triangular number is the sum of the first n natural numbers.
How square numbers are built:
- Start with 1 dot (1 × 1) → 1
- Make a 2 × 2 grid → 4 (add 3 dots to the previous square)
- Make a 3 × 3 grid → 9 (add 5 dots)
- Make a 4 × 4 grid → 16 (add 7 dots)
- Make a 5 × 5 grid → 25 (add 9 dots)
Each time, you add the next odd number. This is why square numbers are sums of consecutive odd numbers.
Connection between triangular and square numbers:
- The sum of two consecutive triangular numbers is always a square number.
- 1 + 3 = 4 = 2²
- 3 + 6 = 9 = 3²
- 6 + 10 = 16 = 4²
- 10 + 15 = 25 = 5²
Types and Properties
Common number patterns in Class 6:
1. Arithmetic patterns (adding the same number)
- 2, 5, 8, 11, 14, ... (adding 3 each time)
- 10, 20, 30, 40, ... (adding 10 each time)
- 3, 6, 12, 24, 48, ... (doubling each time)
- 2, 6, 18, 54, ... (multiplying by 3 each time)
3. Triangular number pattern
- 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
- Differences between consecutive terms: 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
4. Square number pattern
- 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
- Differences between consecutive squares: 3, 5, 7, 9, 11, 13, 15, 17, 19, ...
- The differences are consecutive odd numbers.
5. Patterns with operations
- 1 × 1 = 1
- 11 × 11 = 121
- 111 × 111 = 12,321
- 1,111 × 1,111 = 1,234,321
- The products form palindrome numbers.
- 1 + 2 + 1 = 4 = 2²
- 1 + 2 + 3 + 2 + 1 = 9 = 3²
- 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4²
Solved Examples
Example 1: Example 1: Finding the Next Triangular Number
Problem: The first 5 triangular numbers are 1, 3, 6, 10, 15. Find the 6th and 7th triangular numbers.
Solution:
- 6th triangular number = 6 × 7 / 2 = 42 / 2 = 21
- Or simply: 15 + 6 = 21 (add 6 to the 5th triangular number)
- 7th triangular number = 7 × 8 / 2 = 56 / 2 = 28
- Or: 21 + 7 = 28
Answer: The 6th and 7th triangular numbers are 21 and 28.
Example 2: Example 2: Checking if a Number is a Square Number
Problem: Is 45 a square number? Is 49 a square number?
Solution:
- Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, ...
- 6² = 36 and 7² = 49. So 49 IS a square number.
- 45 is between 36 (6²) and 49 (7²). No whole number squared gives 45.
Answer: 45 is NOT a square number. 49 IS a square number (7²).
Example 3: Example 3: Square Numbers as Sum of Odd Numbers
Problem: Show that 36 is the sum of the first 6 odd numbers.
Solution:
- First 6 odd numbers: 1, 3, 5, 7, 9, 11
- Sum = 1 + 3 + 5 + 7 + 9 + 11 = 36
- And 6² = 36. Confirmed.
Answer: 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²
Example 4: Example 4: Sum of Consecutive Triangular Numbers
Problem: Add the 4th and 5th triangular numbers. What do you get?
Solution:
- 4th triangular number = 10
- 5th triangular number = 15
- Sum = 10 + 15 = 25 = 5²
Answer: 10 + 15 = 25, which is the 5th square number. This confirms: the sum of two consecutive triangular numbers is always a square number.
Example 5: Example 5: Finding the Pattern Rule
Problem: Find the rule and the next two numbers: 5, 11, 17, 23, 29, ...
Solution:
- 11 − 5 = 6
- 17 − 11 = 6
- 23 − 17 = 6
- 29 − 23 = 6
- Rule: Add 6 to get the next number.
- Next numbers: 29 + 6 = 35, 35 + 6 = 41
Answer: The next two numbers are 35 and 41.
Example 6: Example 6: Rectangular Numbers
Problem: Show that 12 is a rectangular number. In how many ways can 12 dots be arranged in a rectangle?
Solution:
A rectangular arrangement means rows × columns, where both rows and columns are at least 2.
- 2 × 6 = 12 (2 rows, 6 columns)
- 3 × 4 = 12 (3 rows, 4 columns)
- 4 × 3 = 12 (same rectangle rotated)
- 6 × 2 = 12 (same rectangle rotated)
Answer: 12 can be arranged as a rectangle in 2 distinct ways: 2 × 6 and 3 × 4.
Example 7: Example 7: Pattern in Multiplication
Problem: Observe the pattern and fill in the blanks: 1 × 8 + 1 = 9, 12 × 8 + 2 = 98, 123 × 8 + 3 = 987, 1234 × 8 + 4 = ?
Solution:
- Following the pattern: 1234 × 8 + 4 = 9,872 + 4 = 9,876
- Check: 1234 × 8 = 9,872. Add 4 = 9,876.
- The answers form the pattern: 9, 98, 987, 9876, ...
Answer: 1234 × 8 + 4 = 9,876
Example 8: Example 8: Sweets in Triangular Pattern
Problem: A sweet shop arranges laddoos in a triangular display. The first row has 1 laddoo, the second row has 2, the third has 3, and so on. If the display has 8 rows, how many laddoos are needed?
Solution:
- Total laddoos = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8
- This is the 8th triangular number = 8 × 9 / 2 = 72 / 2 = 36
Answer: 36 laddoos are needed.
Example 9: Example 9: Dot Pattern for Square Numbers
Problem: How many more dots are needed to go from the 5th square number to the 6th square number?
Solution:
- 5th square number = 5² = 25
- 6th square number = 6² = 36
- Additional dots needed = 36 − 25 = 11
- Notice: 11 is the 6th odd number (1, 3, 5, 7, 9, 11).
Answer: 11 more dots are needed.
Example 10: Example 10: Finding Missing Number in a Pattern
Problem: Complete the pattern: 2, 6, 12, 20, 30, ___, ___
Solution:
Find the differences:
- 6 − 2 = 4
- 12 − 6 = 6
- 20 − 12 = 8
- 30 − 20 = 10
- Differences: 4, 6, 8, 10, ... (increasing by 2 each time)
- Next difference = 12 → 30 + 12 = 42
- Next difference = 14 → 42 + 14 = 56
Answer: The next two numbers are 42 and 56.
(These are actually: 1×2, 2×3, 3×4, 4×5, 5×6, 6×7, 7×8 — products of consecutive numbers.)
Real-World Applications
Real-life uses of number patterns:
- Stacking objects: When you stack oranges or cannonballs in a triangular pyramid shape, the number of objects follows triangular and pyramidal number patterns.
- Seating arrangements: In a cinema with 20 seats in the first row, 22 in the second, 24 in the third, and so on, you use arithmetic patterns to find the total seats.
- Cricket tournaments: In a tournament where every team plays every other team once, the total matches follow triangular numbers. With 8 teams: 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 matches.
- Tiling and flooring: Square tiles arranged in square patterns use square numbers (4, 9, 16, 25, ...).
- Handshakes: If 10 people all shake hands with each other, the total handshakes = 10th triangular number − 10 = 45.
- Savings patterns: If you save ₹10 in week 1, ₹20 in week 2, ₹30 in week 3, and so on, after n weeks you have saved 10 × n(n+1)/2 rupees.
Key Points to Remember
- Triangular numbers are: 1, 3, 6, 10, 15, 21, 28, 36, ... Each is the sum of the first n natural numbers.
- Square numbers are: 1, 4, 9, 16, 25, 36, 49, ... Each is a number multiplied by itself (n²).
- Every square number is the sum of consecutive odd numbers starting from 1.
- The sum of two consecutive triangular numbers is always a square number.
- The difference between consecutive square numbers gives consecutive odd numbers.
- A rectangular number can be arranged in a rectangle with at least 2 rows and 2 columns.
- Every number greater than 1 is either a prime or can be shown as a rectangular number.
- To find the rule in a pattern, look at the differences between consecutive terms.
- If the differences are constant, the pattern is arithmetic (add the same number each time).
- If the differences themselves form a pattern (like 2, 3, 4, 5, ...), look for a deeper rule.
Practice Problems
- Write the first 10 triangular numbers.
- Write the first 10 square numbers.
- Show that 25 is the sum of the first 5 odd numbers.
- Find the 12th triangular number using the formula n(n+1)/2.
- Verify that the sum of the 5th and 6th triangular numbers equals the 6th square number.
- Find the rule and the next three numbers: 1, 4, 9, 16, 25, ___, ___, ___
- A school auditorium has 10 chairs in the first row, 12 in the second, 14 in the third, and so on for 15 rows. How many chairs are in the 15th row?
- Show that 28 is both a triangular number and a rectangular number.
Frequently Asked Questions
Q1. What are triangular numbers?
Triangular numbers are numbers that can be represented by dots arranged in the shape of a triangle. The sequence is 1, 3, 6, 10, 15, 21, 28, ... The nth triangular number is n(n+1)/2.
Q2. What are square numbers?
Square numbers are numbers obtained by multiplying a number by itself. The sequence is 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ... The nth square number is n × n = n².
Q3. Is 1 a triangular number and a square number?
Yes. 1 is the first triangular number (a single dot forms a triangle) and the first square number (1 × 1 = 1). It is both.
Q4. Why is every square number a sum of odd numbers?
When you build a square grid by adding one layer at a time, each new layer adds an odd number of dots. The first square (1×1) has 1 dot. To make 2×2, add 3 dots (one row + one column + corner). To make 3×3, add 5 dots. The pattern of additions is 1, 3, 5, 7, 9, ... — all odd numbers.
Q5. How do I find the rule of a number pattern?
Calculate the differences between consecutive terms. If the differences are the same, the rule is 'add that number.' If the differences form their own pattern (like 2, 4, 6, 8), look for a rule involving multiplication or squares.
Q6. What is the difference between a square number and a rectangular number?
A square number can be arranged in a grid with equal rows and columns (like 3×3 = 9). A rectangular number can be arranged in a grid with unequal rows and columns (like 2×6 = 12). Every square number is also a rectangular number (a square is a special rectangle), but not every rectangular number is a square number.
Q7. Can a number be both triangular and square?
Yes, some numbers are both. The first few are: 1 (1st triangular, 1st square) and 36 (8th triangular, 6th square). The next is 1,225 (49th triangular, 35th square). These are rare.
Q8. What is a rectangular number?
A rectangular number is a number that can be shown as a rectangle with at least 2 rows and 2 columns. For example, 6 = 2×3, 8 = 2×4, 12 = 3×4. Prime numbers cannot be shown as rectangles (they can only be 1×n).
Q9. How many odd numbers do I add to get the 10th square number?
To get the nth square number, add the first n odd numbers. For the 10th square number (100), add the first 10 odd numbers: 1+3+5+7+9+11+13+15+17+19 = 100.
Q10. What is the 100th triangular number?
Use the formula: n(n+1)/2 = 100 × 101 / 2 = 10,100 / 2 = 5,050. The 100th triangular number is 5,050. This is also the sum of the first 100 natural numbers.
