Patterns in Mathematics – Ganita Prakash – Solved

Figure it Out

1. Can you recognise the pattern in each of the sequences in Table 1?

The patterns in each of the sequences from Table 1:

1. All 1’s Sequence:
   • Pattern: Every number in the sequence is 1.
   • Rule: The sequence consists of the same number repeated, i.e., 1, 1, 1, 1, …
2. Counting Numbers:
   • Pattern: The numbers increase by 1 each time.
   • Rule: The sequence is the natural counting numbers, i.e., 1, 2, 3, 4, …
3. Odd Numbers:
   • Pattern: The numbers increase by 2 each time.
   • Rule: These are consecutive odd numbers, i.e., 1, 3, 5, 7, …
4. Even Numbers:
   • Pattern: The numbers increase by 2 each time, starting from 2.
   • Rule: These are consecutive even numbers, i.e., 2, 4, 6, 8, …
5. Triangular Numbers:
   • Pattern: The difference between consecutive terms increases by 1 each time.
   • Rule: These numbers represent triangular arrangements of dots. The nth term is the sum of the first n natural numbers, i.e., 1, 3, 6, 10, 15, …
6. Square Numbers:
   • Pattern: Each number is the square of a natural number.
   • Rule: These are squares of consecutive natural numbers, i.e., 1², 2², 3², 4², … (1, 4, 9, 16, 25, …).
7. Cube Numbers:
   • Pattern: Each number is the cube of a natural number.
   • Rule: These are cubes of consecutive natural numbers, i.e., 1³, 2³, 3³, … (1, 8, 27, 64, 125, …).
8. Virahānka Numbers (Fibonacci Sequence):
   • Pattern: Each number is the sum of the two preceding numbers.
   • Rule: This is the Fibonacci sequence, i.e., 1, 1, 2, 3, 5, 8, 13, …
9. Powers of 2:
   • Pattern: Each number is double the previous number.
   • Rule: These are powers of 2, i.e., 2⁰, 2¹, 2², 2³, … (1, 2, 4, 8, 16, 32, …).
10. Powers of 3:
    • Pattern: Each number is triple the previous number.
    • Rule: These are powers of 3, i.e., 3⁰, 3¹, 3², 3³, … (1, 3, 9, 27, 81, 243, …).

Each of these sequences follows a specific numerical pattern that can be defined with a clear mathematical rule.

2. Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.

Sequence 1:

   

Next three numbers: 1, 1, 1
Rule: This sequence consists entirely of 1’s, so every number in the sequence is always 1.

Sequence 2:

   

Next three numbers: 8, 9, 10
Rule: This is the sequence of counting numbers where each term increases by 1.

Sequence 3:

   

Next three numbers: 15, 17, 19
Rule: This sequence consists of odd numbers where each term increases by 2.

Sequence 4:

   

Next three numbers: 16, 18, 20
Rule: This is the sequence of even numbers, with each term increasing by 2.

Sequence 5:

   

Next three numbers: 36, 45, 55
Rule: These are triangular numbers, where the nth term is the sum of the first n natural numbers.

Sequence 6:

   

Next three numbers: 64, 81, 100
Rule: These are square numbers, where each term is the square of a natural number.

Sequence 7:

   

Next three numbers: 343, 512, 729
Rule: These are cube numbers, where each term is the cube of a natural number.

Sequence 8:

   

Next three numbers: 34, 55, 89
Rule: This is the Fibonacci sequence, where each term is the sum of the two preceding ones.

Sequence 9:

   

Next three numbers: 128, 256, 512
Rule: This is the sequence of powers of 2, where each term is double the previous one.

Sequence 10:

   

Next three numbers: 2187, 6561, 19683
Rule: This is the sequence of powers of 3, where each term is triple the previous one.

Figure it Out

1. Copying the Pictorial Representations: You can copy the pictorial representations from Table 2 into your notebook by sketching the dots and cubes for each sequence. To extend the sequences:

• All 1’s: Just continue with a single dot for each number.
• Counting Numbers: Add an extra dot in a straight line for each number.
• Odd Numbers: Add another dot in the next row to form odd-numbered rows.
• Even Numbers: Continue by adding even-numbered dots in a rectangular shape.
• Triangular Numbers: Add a new row of dots to form the next triangle (21 dots).
• Squares: Add another row and column to form the next square (36 dots).
• Cubes: Draw a cube for the next cube number (216).

2. Why are they called Triangular, Square, and Cube Numbers?:

• Triangular Numbers (1, 3, 6, 10, 15, …) are called so because the dots can be arranged in the shape of a triangle. The nth triangular number is the sum of the first n natural numbers, forming a triangle.
• Square Numbers (1, 4, 9, 16, 25, …) are called square numbers because the dots can be arranged in a perfect square. The nth square number is the square of the natural number n (e.g., 3² = 9).
• Cube Numbers (1, 8, 27, 64, 125, …) are called cubes because they represent the volume of a cube. The nth cube number is the cube of the natural number n (e.g., 2³ = 8).

3. 36 as Both a Triangular and Square Number: You can draw a triangle and a square with 36 dots each:

• For the triangle, arrange the dots in rows, with the first row having 1 dot, the second row 2 dots, and so on until the total number of dots equals 36.
• For the square, arrange the dots in a 6×6 grid to form a square of 36 dots.

4. Hexagonal Numbers:

• Hexagonal numbers are numbers that can be arranged in a hexagon. To draw these, start by placing a central dot, then add layers of dots around it in a hexagonal pattern. The sequence starts as 1, 7, 19, 37, and the next number is 61.

5. Visualizing Powers of 2 and Powers of 3:

• For Powers of 2, you can draw grids where the number of dots doubles with each step (e.g., 1, 2, 4, 8, 16, 32, …).
• For Powers of 3, you can represent them as 3-dimensional cubes, where the number of smaller cubes triples with each step (e.g., 1, 3, 9, 27, 81, …).

And here is a visual representation for Powers of 1, 2, 3, 4, 5: