www.powerpointmaths.com © Where quality comes first!

PowerPointmaths.com © 2004 all rights reserved

Number Sequences

Square NumbersSquare numbers are so called because they can be arranged as a square array of dots.

1

1 x 1= 12

4

2 x 2= 22

9

3 x 3= 32

16

4 x 4= 42

25

5 x 5= 52

36

6 x 6= 62

49

7 x 7= 72

64

8 x 8= 82

81

9 x 9= 92

100

10 x 10= 102

Where do we commonly see Square Numbers?

25

5 x 5= 52

49

7 x 7= 72

100908070605040302010

9081726354453627189

8072645648403224168

7063564942352821147

6054484236302418126

5045403530252015105

403632282420161284

30272421181512963

2018161412108642

10987654321

100816449362516941

S10S9S8S7S6S5S4S3S2S1

Sometimes it’s convenient to use the letter Sn to represent the nth square number, like

below.

S60S50S40S30S20S15S14S13S12S11

Complete the table below for larger square numbers.

S11 S12 S13 S14 S15 S20 S30 S40 S50 S60

121 144 169 196 225 400 900 1600

2500

3600

The rule for the nth square number is simply n2

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

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

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 = T10

551 + 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.

T10

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 = T100

1 + 2 + 3 + ………………… + (n-2) + (n-1) + n

What about in general ( 1)

(1 )2 n

n nSum n T

55453628211510631

T10T9T8T7T6T5T4T3T2T1

Sometimes it’s convenient to use the letter Tn to represent the nth triangular number, like

below.

( 1)2n

n nT

Complete the table below

using the formula

for larger triangular numbers.

T100T50T35T30T25T20T17T15T12T11

Which triangular numbers are also square?1

and 36

505012756304653252101531207866

T100T50T35T30T25T20T17T15T12T11

55453628211510631

T10T9T8T7T6T5T4T3T2T1

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

a2 + b2 = c2

64 36

28

36 + 28 = 64

Number Sequences

Cube NumbersCube numbers can be represented geometrically as a 3 dimensional array of dots or cubes

1

1 x 1 x 1 = 13

8

2 x 2 x 2 = 23

27

3 x 3 x 3 = 33

64

4 x 4 x 4 = 43

125

5 x 5 x 5 = 53

125642781

C10C9C8C7C6C5C4C3C2C1

Sometimes it’s convenient to use the letter Cn to represent the nth cube number, like

below.

Complete the table for the missing cube numbers

1000729512343216125642781

C10C9C8C7C6C5C4C3C2C1

There is a link between sums of cube numbers, square numbers and triangular numbers. Can you figure it out?

+ + +

+ + +

13 = 1

13 + 23 = 9 13 + 23 + 33 = 36

13 + 23 + 33 + 43= 100

1 = 12 = T12

9= 32 = T22

36 = 62 = T32

100 = 102 = T4

2

Sum(13 n3) = Tn2

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

= n2

Pentagonal Numbers:

1 + 4 + 7 + ...+ 3n – 2

= n(3n –1)/2

Hexagonal Numbers:

1 + 5 + 9 + ...+ 4n – 3

= 2n2-nThese 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.

11 The number of reason (the generator of all numbers)

11 The number of opinion (The first female number)

11 The number of harmony (the first proper male number)

11 The number of justice or retribution, indicating the squaring of accounts (Fair and square)

11 The number of marriage (the union of the first male and female numbers)

11 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)