Can You See The Pattern? – CBSE NCERT Study Resources

The given number pattern is 2, 5, 10, 17, 26, ... To identify the rule, let's analyze the differences between consecutive numbers:

The differences themselves form a pattern: 3, 5, 7, 9, ..., which are consecutive odd numbers increasing by 2 each time.

To find the next number:

Following the same rule:

Thus, the next two numbers in the sequence are 37 and 50.

To solve the magic square, let's label the empty cells as follows:

Step 1: Calculate the magic constant (sum of any row, column, or diagonal).

Step 2: Use the third row to find E.

Step 3: Use the middle row to find C.

Step 4: Now, use the second column to find A.

Step 5: Use the third column to find E.

Assuming the magic constant is 15 (common for 3x3 squares with numbers 1-9), let's verify:

Thus, the completed magic square is:

The given number pattern is: 2, 5, 10, 17, 26, ...

(a) The rule followed in this pattern is: Each number is obtained by adding consecutive odd numbers starting from 3 to the previous number.

(b) The next two numbers in the sequence are: 37 and 50.

(c) Here's how the rule is derived:
Start with 2.
Add 3 (the next odd number) to 2: 2 + 3 = 5.
Add 5 (the next odd number) to 5: 5 + 5 = 10.
Add 7 (the next odd number) to 10: 10 + 7 = 17.
Add 9 (the next odd number) to 17: 17 + 9 = 26.
Continuing this pattern:
Add 11 to 26: 26 + 11 = 37.
Add 13 to 37: 37 + 13 = 50.

This pattern is also related to square numbers, as each term is one more than a perfect square (1²+1=2, 2²+1=5, 3²+1=10, etc.).

To solve the magic square, we first need to find the magic constant (the sum for each row, column, and diagonal).

Step 1: Find the magic constant using the completed middle row.
Middle row: _ + 5 + _ = magic constant.
But we can use the first column instead: 8 + _ + 4 = magic constant.
Let's assume the missing number in the first column is 'x'.
8 + x + 4 = 12 + x.

Step 2: Now, look at the main diagonal (top-left to bottom-right): 8 + 5 + _ = magic constant.
This must equal 12 + x (from Step 1).
8 + 5 = 13, so the missing number is (12 + x - 13) = x - 1.

Step 3: Look at the bottom row: 4 + _ + (x - 1) = 12 + x.
4 + y + x - 1 = 12 + x (let y be the middle number in bottom row).
Simplify: 3 + y = 12 → y = 9.

Step 4: Now the square looks like:
8 _ 6
_ 5 _
4 9 _

Step 5: Calculate the magic constant using the bottom row: 4 + 9 + _ = 15 (since 4 + 9 = 13, the last number must be 2).
Now we know the magic constant is 15.

Step 6: Complete the square:
First row: 8 + _ + 6 = 15 → missing number is 1.
First column: 8 + _ + 4 = 15 → missing number is 3.
Main diagonal: 8 + 5 + 2 = 15 (which checks).
Second row: 3 + 5 + 7 = 15 (we find the last missing number is 7).

Final magic square:
8 1 6
3 5 7
4 9 2

We can verify all rows, columns, and diagonals sum to 15.

The given number pattern is 2, 5, 10, 17, 26, ...

(a) The rule used to form this pattern is: Start with 2 and add consecutive odd numbers starting from 3.

(b) The next two numbers in the sequence are 37 and 50.

(c) Here’s how the pattern is derived:

• 2 + 3 = 5
• 5 + 5 = 10
• 10 + 7 = 17
• 17 + 9 = 26
• 26 + 11 = 37
• 37 + 13 = 50

Each step adds the next consecutive odd number (3, 5, 7, 9, ...) to the previous term.

To complete the magic square, we need to ensure the sum of each row, column, and diagonal is equal.

Step 1: Calculate the magic constant (sum of any row).

Using the first row: 8 + _ + 6 = 14 + missing number. Since the center is 5, the magic constant is 3 × 5 = 15 (as the center of a 3x3 magic square is one-third of the magic constant).

Step 2: Fill in the missing numbers.

• First row: 8 + 1 + 6 = 15
• Second row: 9 + 5 + 1 = 15
• Third row: 4 + 9 + 2 = 15

The completed magic square is:

[8, 1, 6]

[9, 5, 1]

[4, 9, 2]

Note: The magic constant is 15, and all rows, columns, and diagonals add up to this value.

The given number pattern is: 2, 5, 10, 17, 26, ...

(i) Rule of the pattern: The sequence is formed by adding consecutive odd numbers starting from 3 to the previous term. Alternatively, it follows the rule: n² + 1, where n is the position of the term (starting from 1).

(ii) Next two numbers: The next two numbers are 37 and 50.

(iii) Explanation: Here's how the pattern works:
1st term: 1² + 1 = 2
2nd term: 2² + 1 = 5
3rd term: 3² + 1 = 10
4th term: 4² + 1 = 17
5th term: 5² + 1 = 26
6th term: 6² + 1 = 37
7th term: 7² + 1 = 50

This shows that each term is the square of its position number plus 1.

Step-by-step solution to complete the magic square:

Given incomplete magic square:
8 | _ | 6
_ | 5 | _
_ | _ | _

Step 1: Find the magic constant (sum of each row/column/diagonal).
We know the center number is 5. In a 3x3 magic square, the magic constant = 3 × center number = 3 × 5 = 15.

Step 2: Fill in the first row.
8 + _ + 6 = 15 → Missing number = 15 - (8 + 6) = 1.
First row becomes: 8 | 1 | 6

Step 3: Fill the main diagonal.
We have two numbers in the main diagonal (8 and 5). The third number can be found by:
8 + 5 + _ = 15 → Missing number = 15 - 13 = 2.
Now the square looks like:
8 | 1 | 6
_ | 5 | _
_ | _ | 2

Step 4: Complete the third column.
6 + _ + 2 = 15 → Missing number = 15 - 8 = 7.
Now we have:
8 | 1 | 6
_ | 5 | 7
_ | _ | 2

Step 5: Complete the second row.
_ + 5 + 7 = 15 → Missing number = 15 - 12 = 3.
Now the square is:
8 | 1 | 6
3 | 5 | 7
_ | _ | 2

Step 6: Complete the first column.
8 + 3 + _ = 15 → Missing number = 15 - 11 = 4.
Now we have:
8 | 1 | 6
3 | 5 | 7
4 | _ | 2

Step 7: Complete the third row.
4 + _ + 2 = 15 → Missing number = 15 - 6 = 9.
Final magic square:
8 | 1 | 6
3 | 5 | 7
4 | 9 | 2

Verification: All rows, columns and diagonals sum to 15.

The given number pattern is 2, 4, 8, 16, 32, ...

(a) The rule used to create this pattern is multiplication by 2. Each number is obtained by multiplying the previous number by 2.
For example:
2 × 2 = 4
4 × 2 = 8
8 × 2 = 16
16 × 2 = 32

(b) The next three numbers in the sequence are:
32 × 2 = 64
64 × 2 = 128
128 × 2 = 256

(c) This pattern is helpful in real-life situations like:

• Calculating the growth of bacteria where they double in number every hour.
• Understanding computer memory sizes (e.g., 2GB, 4GB, 8GB, etc.).
• Planning savings where money doubles over time with compound interest.

The given tiling pattern is: square, triangle, circle, square, triangle, circle, ...

(a) The repeating unit in this pattern is square, triangle, circle. This sequence of three shapes repeats over and over.

(b) The next two shapes in the sequence are:
square (7th shape)
triangle (8th shape)

(c) To find the 10th shape:
The repeating unit has 3 shapes.
10 ÷ 3 = 3 with a remainder of 1.
This means after 3 full cycles (9 shapes), the 10th shape will be the first shape in the repeating unit.
Therefore, the 10th shape is a square.

Here’s how it works:
1. square
2. triangle
3. circle
4. square
5. triangle
6. circle
7. square
8. triangle
9. circle
10. square

The given number pattern is: 2, 5, 10, 17, 26, ...

(i) Rule of the pattern: The sequence is formed by adding consecutive odd numbers starting from 3 to the previous term. Alternatively, it follows the rule: n² + 1, where n is the position of the term (starting from 1).

(ii) Next two numbers:
For the 6th term (n=6): 6² + 1 = 36 + 1 = 37
For the 7th term (n=7): 7² + 1 = 49 + 1 = 50

(iii) Explanation:
Let's verify the pattern:
1st term: 1² + 1 = 2
2nd term: 2² + 1 = 5
3rd term: 3² + 1 = 10
4th term: 4² + 1 = 17
5th term: 5² + 1 = 26
This confirms the rule works perfectly!

(i) Lines of symmetry: The original figure (square) has 4 lines of symmetry – two diagonals and one vertical and one horizontal line passing through the center.

(ii) Shading for symmetry: To maintain symmetry after shading the top-left square, you must shade its mirror image. For example:

• If shaded using the vertical line of symmetry, shade the top-right square.
• If shaded using the horizontal line, shade the bottom-left square.
• If shaded using the diagonal (top-left to bottom-right), shade the bottom-right square.

(iii) Alternate symmetrical figure: Imagine a circle divided into 6 equal slices (like a pie chart). Shading one slice would mean its symmetrical counterparts (depending on the line of symmetry) would also be shaded. For example, with a vertical line of symmetry, shading the rightmost slice would require shading the leftmost slice too.

The given number pattern is 2, 4, 8, 16, 32, ...

a) Rule of the pattern: Each number is obtained by multiplying the previous number by 2. This is called a geometric pattern where the ratio between consecutive terms is constant.

b) Next three numbers: Following the rule, the next three numbers are:
32 × 2 = 64
64 × 2 = 128
128 × 2 = 256

c) Representation using multiplication: The pattern can be written as powers of 2:
2 = 2¹
4 = 2²
8 = 2³
16 = 2⁴
32 = 2⁵
This shows exponential growth, which is faster than addition-based patterns.

a) Next three shapes: The pattern repeats as Square, Triangle, Square, Triangle, ...
So, the next three shapes after the given sequence are:
1. Square
2. Triangle
3. Square

b) 10th shape: Since the pattern alternates every term:
Odd positions (1st, 3rd, 5th, ...) = Square
Even positions (2nd, 4th, 6th, ...) = Triangle
10 is an even number, so the 10th shape is a Triangle.

c) Real-life application: Such alternating patterns are used in:

• Floor tiles to create visually appealing designs.
• Textile prints for clothes and curtains.
• Architectural layouts to balance symmetry and variety.

a) Matchsticks for Figure 5:
Figure 1: 3 matchsticks
Figure 2: 3 + 3 = 6 matchsticks
Figure 3: 6 + 3 = 9 matchsticks
The pattern increases by 3 matchsticks per figure.
So, Figure 4: 9 + 3 = 12 matchsticks
Figure 5: 12 + 3 = 15 matchsticks.

b) General rule: For Figure n, the number of matchsticks = 3 × n.
This is because each new triangle adds 3 matchsticks to the previous total.

c) Other closed shapes: Yes, this pattern can form:

• A hexagon by arranging 6 triangles (18 matchsticks).
• A star by overlapping triangles.