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3. Number Sequences Square Numbers Square numbers are so called because they can be arranged as a square array of dots. 1 1 x 1 = 1 2 4 2 x 2 = 2 2 9 3 x 3 = 3 2 16 4 x 4 = 4 2 25 5 x 5 = 5 2 36 6 x 6 = 6 2 49 7 x 7 = 7 2 64 8 x 8 = 8 2 81 9 x 9 = 9 2 100 10 x 10 = 10 2

4. Where do we commonly see Square Numbers? 25 5 x 5 = 5 2 49 7 x 7 = 7 2 100816449362516941 S 10 S9S9 S8S8 S7S7 S6S6 S5S5 S4S4 S3S3 S2S2 S1S1 Sometimes it’s convenient to use the letter S n to represent the nth square number, like below. S 60 S 50 S 40 S 30 S 20 S 15 S 14 S 13 S 12 S 11 Complete the table below for larger square numbers. S 11 S 12 S 13 S 14 S 15 S 20 S 30 S 40 S 50 S 60 1211441691962254009001600 25003600 The rule for the nth square number is simply n 2

5. Number Sequences Triangular Numbers Triangular numbers are so called because they can be arranged in a triangular array of dots. 1 1 3 1 + 2 6 1 + 2 + 3 10 1 + 2 + 3 + 4 15 1 + 2 + 3 + 4 + 5 21 1 + 2 + 3 + 4 + 5 + 6 28 1 + 2 + 3 + 4 + 5 + 6 + 7 36 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 45 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 55 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 To find the nth triangular number you simply add up all the numbers from 1 to n

6. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 Adding in pairs gives: sum of the numbers from (1  10) = 5 x 11 = 55 = T 10 55 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 If you don’t know the rule for this then there is a clue below that should help you figure out a method for the numbers 1 to 10. T 10 This method of adding in pairs can be used to add up any set of consecutive whole numbers from (1  n). What do the numbers from 1 to 100 add up to? 1 + 2 + 3 + ………………… + 98 + 99 + 100 50 x 101 = 5050 = T 100 1 + 2 + 3 + ………………… + (n-2) + (n-1) + n What about in general

7. 55453628211510631 T 10 T9T9 T8T8 T7T7 T6T6 T5T5 T4T4 T3T3 T2T2 T1T1 Sometimes it’s convenient to use the letter T n to represent the nth triangular number, like below. Complete the table below using the formula  for larger triangular numbers. T 100 T 50 T 35 T 30 T 25 T 20 T 17 T 15 T 12 T 11 Which triangular numbers are also square? 1 and 36 50501275 6304653252101531207866 T 100 T 50 T 35 T 30 T 25 T 20 T 17 T 15 T 12 T 11

8. 55453628211510631 T 10 T9T9 T8T8 T7T7 T6T6 T5T5 T4T4 T3T3 T2T2 T1T1 Look at the table of triangular numbers below. Can you find a link to square numbers Any pair of adjacent triangular numbers add to a square number 1 + 3 = 4 3 + 6 = 9 6 + 10 = 16 45 + 55 = 100 The followers of Pythagoras in ancient Greece were the first people to discover this relationship. By drawing a single straight line on the diagram below can you see why this is. Pythagoras (570-500 b.c.) c a b a 2 + b 2 = c 2 64 36 28 36 + 28 = 64

9. Number Sequences Cube Numbers Cube numbers can be represented geometrically as a 3 dimensional array of dots or cubes 1 1 x 1 x 1 = 1 3 8 2 x 2 x 2 = 2 3 27 3 x 3 x 3 = 3 3 64 4 x 4 x 4 = 4 3 125 5 x 5 x 5 = 5 3 125642781 C 10 C9C9 C8C8 C7C7 C6C6 C5C5 C4C4 C3C3 C2C2 C1C1 Sometimes it’s convenient to use the letter C n to represent the nth cube number, like below. Complete the table for the missing cube numbers 1000 729512343216125642781 C 10 C9C9 C8C8 C7C7 C6C6 C5C5 C4C4 C3C3 C2C2 C1C1

10. There is a link between sums of cube numbers, square numbers and triangular numbers. Can you figure it out? + + + + + + 1 3 = 1 1 3 + 2 3 = 9 1 3 + 2 3 + 3 3 = 36 1 3 + 2 3 + 3 3 + 4 3 = 100 1 = 1 2 = T 1 2 9= 3 2 = T 2 2 36 = 6 2 = T 3 2 100 = 10 2 = T 4 2 Sum(1 3  n 3 ) = T n 2

11. Pythagoras and his followers discovered many patterns and relationships between whole numbers. Triangular Numbers: 1 + 2 + 3 +...+ n = n(n + 1)/2 Square Numbers: 1 + 3 + 5 +...+ 2n – 1 = n 2 Pentagonal Numbers: 1 + 4 + 7 +...+ 3n – 2 = n(3n –1)/2 Hexagonal Numbers: 1 + 5 + 9 +...+ 4n – 3 = 2n 2 -n These figurate numbers were extended into 3 dimensional space and became polyhedral numbers. They also studied the properties of many other types of number such as Abundant, Defective, Perfect and Amicable. In Pythagorean numerology numbers were assigned characteristics or attributes. Odd numbers were regarded as male and even numbers as female. 1.  The number of reason (the generator of all numbers) 2.  The number of opinion (The first female number) 3.  The number of harmony (the first proper male number) 4.  The number of justice or retribution, indicating the squaring of accounts (Fair and square) 5.  The number of marriage (the union of the first male and female numbers) 6.  The number of creation (male + female + 1) 10.  The number of the Universe (The tetractys. The most important of all numbers representing the sum of all possible geometric dimensions. 1 point + 2 points (line) + 3 points (surface) + 4 points (plane)