All questions of Patterns in Mathematics for Class 6 Exam

64 is the square of 8
8×8='64'

This sequence represents square numbers (1x1, 2x2, 3x3, 4x4...). The next number is 5x5, which equals 25.

A square number is a number that can be represented as the product of an integer multiplied by itself. 25 is a square number because it is equal to 5x5.

The first three odd numbers are 1, 3, and 5. Adding them together gives 1 + 3 + 5 = 9.

The sequence 3, 4, 5, 6 represents the number of sides in regular polygons like triangles, squares, pentagons, and hexagons.

Understanding Triangular Numbers
Triangular numbers are formed by the sum of the natural numbers. The nth triangular number is given by the formula:
- T(n) = n(n + 1)/2
For example:
- T(1) = 1
- T(2) = 3
- T(3) = 6
- T(4) = 10
- T(5) = 15
- T(6) = 21
- T(7) = 28
- T(8) = 36
Understanding Square Numbers
Square numbers are the result of multiplying a number by itself. The nth square number is given by the formula:
- S(n) = n^2
For example:
- S(1) = 1
- S(2) = 4
- S(3) = 9
- S(4) = 16
- S(5) = 25
- S(6) = 36
Identifying the Correct Option
Now, let's evaluate the options to see which one is both a triangular and a square number.
- Option a) 25: Not a triangular number (T(6) = 21, T(7) = 28)
- Option b) 36:
- Triangular: T(8) = 36
- Square: S(6) = 36
- Option c) 8: Not a triangular or square number
- Option d) 4: Square (S(2) = 4) but not triangular (T(2) = 3)
Conclusion
The only number that is both a triangular number and a square number is 36. Hence, the correct answer is option b.

Adding 2 repeatedly to the number 1 creates a sequence of odd numbers (1, 3, 5, 7, and so on). This is because odd numbers are separated by a difference of 2.

By adding the first few odd numbers (1 + 3 = 4, 1 + 3 + 5 = 9, and so on), you get square numbers like 1, 4, 9, and 16.

A cube is a three-dimensional object with equal sides and angles in all directions, while the other shapes are two-dimensional.

Understanding the Sequence
The given sequence is 1, 3, 6, __, 15. To find the missing term, let's analyze the pattern between the numbers.
Identifying the Pattern
- The first term is 1.
- The second term (3) can be derived from the first term by adding 2:
1 + 2 = 3
- The third term (6) can be derived from the second term by adding 3:
3 + 3 = 6
Finding the Missing Term
Now, we need to find the missing term, which we will denote as x. Following the established pattern:
- The next step involves adding 4 to the previous term:
6 + 4 = x
Therefore, x = 10.
Continuing the Pattern
- To verify, we check the next term (15):
- Adding 5 to the missing term:
10 + 5 = 15
This confirms that our missing term is correct.
Conclusion
Thus, the missing term in the sequence is:
- 10 (option 'C').
This sequence is an example of a pattern where each term increases by an incrementally larger number (2, 3, 4, 5, ...), resulting in 10 being the right answer.

The third triangular number has 6 dots, forming a triangle with 3 rows.

Adding 2 repeatedly to the number 2 creates a sequence of even numbers (2, 4, 6, 8, and so on). This is because even numbers are separated by a difference of 2.

Understanding Hexagons
A hexagon is a polygon that is defined by its number of sides. To understand why a hexagon has six sides, let’s break it down:
Definition of a Hexagon
- A hexagon is a two-dimensional shape.
- It consists of straight lines that connect to form a closed figure.
Counting the Sides
- The prefix "hex" comes from the Greek word for six.
- Therefore, a hexagon must have exactly six straight sides.
Visual Representation
- Imagine drawing a shape with six equal-length straight lines.
- When you join the ends of these lines, you create a closed figure with six angles.
Real-Life Examples
- Honeycombs are a natural example of hexagonal shapes.
- Many tiles and patterns in art and architecture feature hexagons due to their efficiency in space and aesthetic appeal.
Conclusion
- In conclusion, a hexagon is defined by its six sides.
- Therefore, the correct answer to the question of how many sides a hexagon has is option 'b), 6'.
Understanding the characteristics of polygons, such as hexagons, helps in grasping fundamental geometry concepts that are essential in higher-level mathematics.

A square is a regular polygon because it has equal-length sides and equal angles. The other shapes do not meet both criteria.

The sequence 1, 3, 6, 10, 15 represents triangular numbers, which can be visualized as dots forming triangles.

36 is both a square number (6x6) and a triangular number, making it unique in this context.

Regular polygons and number sequences both have patterns. In polygons, the pattern is in the sides and angles, while in number sequences, the pattern is in the numbers themselves.

The sequence 3, 6, 9, 12 follows the pattern of multiplying by 3. This creates a sequence where each number is a multiple of 3.