Transcript Number Patterns - Sunderland Learning Hub

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Number Sequences
Square Numbers
Square numbers are so called because they can be arranged as a square array of dots.
4
1
1 x 1
= 12
49
7 x 7
= 72
2 x 2
= 22
9
16
3 x 3
= 32
4 x 4
= 42
64
8 x 8
= 82
25
5 x 5
= 52
81
9 x 9
= 92
36
6 x 6
= 62
100
10 x 10
= 102
Where do we commonly see Square Numbers?
25
5 x 5
= 52
1
2
3
4
5
6
7
8
9
10
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9
18
27
36
45
54
63
72
81
90
10
20
30
40
50
60
70
80
90
100
49
7 x 7
= 72
Sometimes it’s convenient to use the letter Sn to represent the nth square number, like below.
Complete the
table below
for larger
square
numbers.
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
1
4
9
16
25
36
49
64
81
100
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
28
1+2+3+4+5+6+7
3
6
1+2
1+2+3
10
1+2+3+4
36
1+2+3+4+5+6+7+8
21
15
1+2+3+4+5
1+2+3+4+5+6
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
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.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
55
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
T10
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
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
What about in general
1 + 2 + 3 + ………………… + (n-2) + (n-1) + n
Sum (1  n ) 
n (n  1)
2
 Tn
Sometimes it’s convenient to use the letter Tn to represent the nth triangular number, like below.
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
1
3
6
10
15
21
28
36
45
55
1
and
36
Which triangular numbers are also square?
Complete the table below
using the formula
Tn 

for larger triangular numbers.
T17
T20
T25
T30
n (n  1)
2
T11
T12
T15
T35
T50
T100
66
78
120 153 210 325 465 630
1275
5050
Look at the table of triangular numbers below. Can you find a link to square numbers
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
1
3
6
10
15
21
28
36
45
55
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.)
64
36
c
36 + 28 = 64
28
a
a2 + b2 = c2
b
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 = 13
27
8
3 x 3 x 3 = 33
2 x 2 x 2 = 23
125
64
5 x 5 x 5 = 53
4 x 4 x 4 = 43
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
C1
C2
C3
C4
1
8
27
64
C5
C6
C7
C8
C9
125 216 343 512 729
C10
1000
There is a link between sums of cube numbers, square
numbers and triangular numbers. Can you figure it out?
+
+
13 + 23 = 9
13 = 1
+
13 + 23 + 33 = 36
1 = 12 = T12
9= 32 = T22
+
+
+
36 = 62 = T32
100 = 102 = T42
13 + 23 + 33 + 43= 100
Sum(13  n3) = Tn2
Pythagoras and his followers discovered many patterns and relationships between whole numbers.
Triangular Numbers:
Square Numbers:
Pentagonal Numbers:
Hexagonal Numbers:
1 + 2 + 3 + ...+ n
1 + 3 + 5 + ...+ 2n – 1
1 + 4 + 7 + ...+ 3n – 2
1 + 5 + 9 + ...+ 4n – 3
= n(n + 1)/2
= n2
= n(3n –1)/2
= 2n2-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)