Number Patterns
Geometric
𝑇
= 𝑇× 𝑟
𝑂𝑅
𝑇
= 𝑇× 𝑟
Lineararithmetic
𝑇
= 𝑇+ 𝑛 − 1 𝑑
𝑇
= 𝑇+ 𝑛𝑑
𝑂𝑅
First differences are constant
Each sucessive term is found by adding
a constant value to the previous term.
Constant value being added each time in
a specific pattern is termed "the
difference" for that pattern and denoted
by
d
Quadratic arithmetic
𝑇
= 𝑎𝑛+ 𝑏𝑛 + 𝑐
Second differences are constant
The values being added to terms to find successive terms are
no longer Constant. Instead these values form a linear
arithmetic pattern in their own right.
𝒂
+
𝒃
+
𝒄
4
𝑎
+
2
𝑏
+
𝑐
9
𝑎
+
3
𝑏
+
𝑐
𝟑𝒂
+
𝒃
5
𝑎
+
𝑏
𝟐𝒂
𝑇
𝑇
𝑇
𝑇
−
𝑇
𝑇
−
𝑇
first differences - not
constant - forms a
linear pattern
constant second
difference
IMPORTANT: The list of square numbers is the most basi c quadratic pattern
𝑇
= 𝑛 ⟹ 1 ; 4 ; 9 ;16 ; 25 ; 36 ; 49 ; 64 ;81 ; 100 ; 121 ; 144 ;
Arithmetic
The set pattern between terms involve adding
specific values to the previous term
Cubic arithmetic
Third differences are constant
The values being added to terms to find successive terms are
now from a quadratic arithmetic pattern.
IMPORTANT: The list of cube numbers is the most basic cubic pattern
𝑇
= 𝑛 ⟹ 1 ; 8 ; 27;64 ;125 ;216 ;343 ;512 ;
𝑇
𝒂
+
𝒃
+
𝒄
+
𝒅
8
𝑎
+
4
𝑏
+
2
𝑐
+
𝑑
27
𝑎
+
9
𝑏
+
3
𝑐
+
𝑑
𝟕𝒂
+
𝟑𝒃
+
𝒄
19
𝑎
+
5
𝑏
+
𝑐
𝟏𝟐𝒂
+
𝟐𝒃
64𝑎 + 16𝑏 + 4𝑐 + 𝑑
37𝑎 + 7𝑏 + 𝑐
18𝑎 + 2𝑏
𝟔𝒂
𝑇
𝑇
𝑇
first differences - not
constant - forms a
quadratic pattern
second differences - not constant -
forms a linear pattern
constant third difference
𝑇
= 𝑎𝑛+ 𝑏𝑛+ 𝑐𝑛 + 𝑑
Not the same "d" as in linear patterns
Constant ratio
Successive terms are found by multiplying
the previous term by a constant value, termed
"the ratio" and denoted by
r
A list of numbers that follow a set pattern in moving
from one term to the next term in the list
0123456
25 8 11 14 17 20
𝑛
𝑇
n
is used to denote the position of a term in the pattern
Always STARTS AT 1 and can only be INTEGER values
Tnis used to denote the actual terms of the pattern
T3= 11 11 is the value of the 3rd term
A purely theoretical term of the pattern - patterns always start at a
first term and there is no such thing as a zeroth term.
BUT the value of this theoretical term is very useful
also called
Exponential patterns
Geometric patterns often occur in real life scenarios
Compound interest on investments or loans, depreciation of assets, inflation in cost of everyday goods, population growth and decay - all involve geometric patterns.
In these patterns the ratio's are always time based - m eaning that nis directly linked to the number of time periods that have lapsed, but NOT necessarily equal)
When we model these situations using geometric pat terns, we prefer to view the starting value as the zeroth term (T0formula)instead of the first term- t his eliminates the
potential mismatch between the value of nand t he number of lapsed time periods
n 1 2 3 4 5 6 7
$
100 110 121 133 146 161 177
n 0 1 2 3 4 5 6
A $100 is invested at
10% per year compound
interest
⟹ 𝟓𝒕𝒉 𝑡𝑒𝑟𝑚 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑎𝑓𝑡𝑒𝑟 𝟒 𝒚𝒆𝒂𝒓𝒔
$
=
100
(
1
.
1
)
𝑛
𝑖𝑠
𝑁𝑂𝑇
𝑡
ℎ
𝑒
𝑛𝑢𝑚𝑏𝑒𝑟
𝑜𝑓
𝑦𝑒𝑎𝑟𝑠
⟹ 𝟒𝒕𝒉 𝑡𝑒𝑟𝑚 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑎𝑓𝑡𝑒𝑟 𝟒 𝒚𝒆𝒂𝒓𝒔
$
=
100
(
1
.
1
)
𝑛
𝐼𝑆
𝑡
ℎ
𝑒
𝑛𝑢𝑚𝑏𝑒𝑟
𝑜𝑓
𝑦𝑒𝑎𝑟𝑠
The Tnequation is often called the nth
term or general term rule/formula
Shortcut for finding the general rule for certain quadratic and cubic arithmetic patterns
If a given pattern can be closely matched to the known basic quadratic or cubic patterns, the general
rule for it can be easily deduced
All our terms are -10 from
sequential terms in the basic
cubic pattern
54 115 206 333 …
−
10
−
10
−
10
−
10
𝑇
= 𝑛 ⟹
1
;
8
;
27
;
64
;
125
;
216
;
343
;
512
;
Our first term is 3 positions away from the original first term
𝑻
𝒏
=
(
𝒏
+
𝟑
)
𝟑
−
𝟏𝟎
