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Prime Factorisation:


The Prime Factorisation of a number acts like its finger prints.
It helps to identify the prime factors that multiply to make the number and can aid in solving the highest common factor (HCF) and lowest common multiple (LCM)To get the prime factorisation we use a factor tree or a factor column Example:
108 = 2² x 3³

Factors and Multiples:


Highest Common Factor
To find the highest common factor, you multiply the same prime factors in two numbers.
Example:
36 and 44
36 = 2² x 3²
44 = 2² x 11
Therefore the highest common factor of 36 and 44 is 2² as it occurs in both numbers.

Lowest Common Multiple


To find the lowest common multiple, you multiply the all the highest form of all the prime factors.
Example:
36 and 270
36 = 2² x 3²
270 = 2 x 3³ x 5
Therefore the lowest common multiple, you multiply 2² x 3³ x 5 = 540.


Linear Sequences:


A linear sequence is a sequence which has the same difference between two consecutive terms.
The formula is found to determine the term relative to its position in the sequence.
Hence it is called the 'position to term' rule.
It is found by the formula SNOT.
Step x N + 0th Term
For example:
In a sequence: 4, 9, 14, 19, 24 ...
It is solved by being put in a table like this:

Position (n)
0
1
2
3
4
5
Term
-1
4
9
14
19
24

Step: +5
0th Term: -1
Therefore the position to term rule for his linear sequence is 5n - 1



Arithmetic Series (Progressions)
The sum to n terms of an arithmetic sequence (Sn):
General form: T₁+ T₂+ T₃+ T₄....+Tn (not T x n)
a = 1st term = T
d = common difference
(a+ (a+d) + (a+2d) (a+3d) +...(a+(n-1)d)

Now:
l = last term = Tn
n = the no. of terms
Eqaution 1
Sn = a + a + d + a + 2d + .... + l - 2d + l - d + l

Also the sequence could be written in reverse
Eqaution 2
Sn = l + l - d + l - 2d .... + a + 2d + a + d + a

Equation 1 + 2
2Sn = (a + l) + (a + l) + (a + l) + ..... (a + l) + (a + l)
= (a + l) .... n sets

2Sn = n(a + l)

Therefore: Sn = n/2(a + l)

Since there are n terms, l = Tn = a + (n-1)d
So substitute » Sn = n/2(a + a + (n - 1)d)

Therefore: Sn = n/2(2a + (n - 1)d)

Geometric Sequences

A geometric sequence is one in which there is a common ratio between consecutive terms.

Property: T₂/T₁= T₃/T₂= T₄/T₃.... = Tn/T(n-1) = r the common ratio

General form of a geometric sequence:
a; ar; ar² ....

General Term (Tn):
T₁= a = ar⁰ = ar¹⁻¹
T₂= ar¹ = ar²⁻¹
T₃= ar² = ar³⁻¹

Therefore: Tn = ar^n-1

By Muhammed