All questions of Patterns in Mathematics for Class 6 Exam

64 is the square of 8
8×8='64'

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.

Because other words  are not related to mathematics. Words, colours and sounds are related to Science.

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.

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

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.

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.

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

Understanding Triangular Numbers
Triangular numbers are a sequence of numbers that can form an equilateral triangle. The nth triangular number is the total number of dots that can be arranged in a triangle with n rows.

Formula for Triangular Numbers
To find the nth triangular number, we use the formula:

• T(n) = n(n + 1) / 2

Here, T(n) represents the nth triangular number.

Calculating the Third Triangular Number
To find the third triangular number (n = 3):

• T(3) = 3(3 + 1) / 2
• T(3) = 3(4) / 2
• T(3) = 12 / 2
• T(3) = 6

So, the third triangular number is 6.

Visualizing the Third Triangular Number
You can visualize the arrangement of dots as follows:

• First row: 1 dot
• Second row: 2 dots
• Third row: 3 dots

When you add these up:

• 1 + 2 + 3 = 6

This confirms that the total number of dots in the third triangular number is indeed 6.

Conclusion
Thus, the correct answer to the question about the number of dots in the third triangular number is option 'B', which is 6.

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

Regular Polygons and Number Sequences:
Regular polygons and number sequences have a commonality in the sense that both exhibit patterns. Let's delve into the details:

Regular Polygons:
- Regular polygons are two-dimensional shapes with equal sides and angles.
- They follow a specific pattern in terms of the number of sides and angles they possess.
- For example, a triangle has 3 sides and 3 angles, a square has 4 sides and 4 angles, and so on.
- The pattern in regular polygons lies in the relationship between the number of sides and the interior angles.

Number Sequences:
- Number sequences are ordered lists of numbers that follow a specific pattern or rule.
- They can be arithmetic, geometric, Fibonacci, etc., each with its own pattern.
- For example, in an arithmetic sequence, each term is obtained by adding a constant value to the previous term.
- In a geometric sequence, each term is obtained by multiplying the previous term by a constant value.

Commonality - Patterns:
- The commonality between regular polygons and number sequences is that both exhibit patterns.
- Regular polygons follow a pattern in the number of sides and angles they possess.
- Number sequences follow a pattern in the way each term is derived from the previous term.
- Recognizing and understanding these patterns is crucial in solving problems related to both regular polygons and number sequences.

Understanding the Sequence
The given sequence is: 1, 4, 9, 16...
Each number in this sequence is a perfect square.
Identifying the Pattern
- The first number (1) is \(1^2\).
- The second number (4) is \(2^2\).
- The third number (9) is \(3^2\).
- The fourth number (16) is \(4^2\).
Thus, the pattern is that the \(n\)-th term in the sequence is given by \(n^2\).
Finding the Next Number
To find the next number in the sequence, we need to calculate \(5^2\):
- \(5^2 = 25\)
Conclusion
The next number in the sequence is 25, which corresponds to option 'C'.
- This confirms that the sequence consists of the squares of consecutive integers: \(1^2, 2^2, 3^2, 4^2, 5^2\).
Understanding this pattern is crucial for solving similar problems in the future.

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.

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

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

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

A hexagon is a regular polygon with 6 equal sides and angles.