• Patterns occur when a sequence of numbers obeys a certain rule. For example:
  • 1, 3, 6, 10, 15, … are triangular numbers.
  • 1, 4, 9, 16, 25, … are square numbers.
  • 1, 1, 2, 3, 5, 8, 13, … is a Fibonacci sequence.
  • 3, 6, 9, 12, 15, … are multiples of 3.
• Number patterns can be shown in a diagram or in a simple graph.
• If a × a = b, then b is the square of a, and a is the square root of b. For example:
  • Since 4 × 4 = 16, 16 is the square of 4, and 4 is the square root of 16.
• If the square root of a number, p, is a whole number, then p is a perfect square. For example:
  • 121 is a perfect square because its square root is 11 (a whole number).

Examples

1. Find the next four terms in the sequence 1, 2, 4, 7, 11, 16, … (Method: Find the difference between two consecutive numbers)
   • 
   • Notice the pattern in the differences. The differences increase by 1 each time.
   • The next term in the sequence is found by adding 6 to 16, which gives 22.
   • The next term is found by adding 7 to 22, and so on.
   • ∴ The next four terms are: 22, 29, 37, and 46.
2. Find √11025. (Hint: Try the prime numbers 2, 3, 5 ,7, … in turn)
   • Working:
     
   • 11025 = 3² × 5² × 7²
   •           = (3 × 5 × 7) × (3 × 5 × 7)
   •           = 105 × 105
   • ∴ √11025 = 105
   • NOTE: It is not necessary to write a number in its prime factors.
3. Find √6400
   • 6400 = 64 × 100
   •          = 8² × 10²
   • Thus = √6400 = 8 × 10 = 80
4. Find the smallest number which 540 must be multiplied so that the product is a perfect square.
   

• 540 = 2² × 3² × 5
• The index of 2 is even.
• The indices of 3, and 5 are odd.
• One more 3 and one more 5 will make all the indices even. the product will then be a perfect square.
• The number required = 3 × 5 = 15
• What are the next two values in the following sequence?
  1, 8, 27, 64, 125, 216, …
  • If you check, the first term is the cube of 1; 1³ = 1.
  • The second term is the cube of 2; 2³ = 8.
  • The third term is the cube of 3; 3³ = 27.
  • ∴ The seventh term would be 7³ = 343.
  • ∴ The eighth term would be 8³ = 512
• What are the next two values in the following sequence?
  15, 17, 20, 24, 29, 35, …
  • The difference between the first and second term is 2; 17 – 15 = 2.
  • The difference between the third and second term is 3; 20 – 17 = 3.
  • The difference between the fourth and third term is 4; 24 – 20 = 4.
  • ∴ The difference between the seventh and sixth term has to be 7:
    • ∴ x – 35 = 7
    • ∴ x = 42
    • ∴ The seventh term is 42.
  • ∴ The difference between the eighth and seventh term has to be 8:
    • ∴ y – 42 = 8
    • ∴ y = 50
    • ∴ The eighth term is 50.