SEQUENCE

Sequences

Arithmetic Sequences

An Arithmetic Sequence is made by adding the same value each time. The value added each time is called the "common difference"

Example 1:

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time, like this:

Example 2:

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.
The pattern is continued by adding 5 to the last number each time, like this:

The common difference could also be negative.

Example 3:

25, 23, 21, 19, 17, 15, ...

This common difference is −2
The pattern is continued by subtracting 2 each time, like this:

Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin 

When a coin is tossed, there are two possible outcomes:

• heads (H) or
• tails (T)

We say that the probability of the coin landing H is ½.

And the probability of the coin landing T is ½.

Example 1:

The chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 16

Example 2:

There are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 45 = 0.8

Example 3:

There are 7 marbles in a bag: 5 are blue, and 2 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 2 (there are 2 blues)

Total number of outcomes: 7 (there are 7 marbles in total)

So the probability = 27

Exercise:

There are 6 flowers; 1 lavender, 2 roses and 3 sunflower. What is the probability of getting

(a) Lavender roses

(b) sunflower roses

INEQUALITIES

Introduction to Inequalities

Inequality tells us about the relative size of two values.

Mathematics is not always about "equals"! Sometimes we only know that something is bigger or smaller.

Greater or Less Than

The two most common inequalities are:

Symbol	Words	Example Use
>	greater than	5 > 2
<	less than	7 < 9

... Or Equal To!

We can also have inequalities that include "equals", like:

Symbol	Words	Example Use
≥	greater than or equal to	x ≥ 1
≤	less than or equal to	y ≤ 3

Example 1:

Alex and Billy have a race, and Billy wins!

What do we know?

We don't know how fast they ran, but we do know that Billy was faster than Alex:

Billy was faster than Alex

We can write that down like this:

b > a

(Where "b" means how fast Billy was, ">" means "greater than", and "a" means how fast Alex was).

We call things like that inequalities (because they are not "equal").

Example 2:

Alex plays in the under 15's soccer. How old is Alex?

We don't know exactly how old Alex is, because it doesn't say "equals"

But we do know "less than 15", so we can write:

Age < 15

The small end points to "Age" because the age is smaller than 15.

Example 3 is equal to:

You must be 13 or older to watch a movie.

The "inequality" is between your age and the age of 13.

Your age must be "greater than or equal to 13", which is written:

Age ≥ 13

Exercise:

Represent the following sentence into inequalities:-

a) To watch a parental guidance movie, you must be 18 years and above.

b) Ali is twice older than Hisyam

LOGARITHM

Introduction to Logarithm

In its simplest form, a logarithm answers the question:

How many of one number do we multiply to get another number?

Example 1: 

How many 2s do we multiply to get 8?

Answer 

2 x 2 x 2 = 8 

so we needed to multiply 3 of the 2s to get 8.

So the logarithm is 3. 

How to Write it:

We write "the number of 2s we need to multiply to get 8 is 3" as:

log₂(8) = 3

Example 2:

What is log₅(625) ... ?

We are asking "how many 5s need to be multiplied together to get 625?"

5 × 5 × 5 × 5 = 625, so we need 4 of the 5s

Answer: log₅(625) = 4

Example 3:

What is log₂(64) ... ?

We are asking "how many 2s need to be multiplied together to get 64?"

2 × 2 × 2 × 2 × 2 × 2 = 64, so we need 6 of the 2s

Answer: log₂(64) = 6

Exercise:

What is log3(27) ... ?

MEAN

What is Mean and how do we find it?

The mean is the average of the numbers.

It is easy to calculate: add up all the numbers, then divide by how many numbers there are.

In other words it is the sum divided by the count.


Example 1:

What is the Mean of these numbers?

6, 11, 7

• Add the numbers: 6 + 11 + 7 = 24
• Divide by how many numbers (there are 3 numbers): 24 / 3 = 8

The Mean is 8.

Example 2:

Look at these numbers: 

3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

The sum of these numbers is 330

There are fifteen numbers.

The mean is equal to 330 / 15 = 22

The mean of the above numbers is 22.

Example 3:

Look at these numbers: 

5, 4, 3, 12, 2

The sum of these numbers is 26

There are five numbers.

The mean is equal to 26 / 5 = 5.5

The mean of the above numbers is 5.5.

Exercise:

Look at these numbers: 

12, 12, 14, 15, 23, 24, 25

Calculate the mean of the above numbers

INDICES

INDICES

Indices is the index of a number that says how many times to use the number in a multiplication. 

Rules of indices:

Example 1:

Work out 2^5 x 2^6 = 2^(5+6) = 2^11


Example 2:

Work out (4^5)^2 = 4^10



Example 3:

4/4 = 1 , 7/7 = 1 and 40/40 = 1

Exercise
a) Work out (4^2)^7
b) Work out 3^2 x 3^5