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Number Pattern Formula for Arithmetic Sequences: Tn = a + (n – 1)d, where n is the ordinal numerical value of the term, a is the first term and d is the common difference between any two consecutive terms.
Geometric Sequences
Number Pattern Formula for Geometric Sequence: Tn = arn–1 where n is the ordinal numerical value of the term, a is the first term and r is the common ratio between any two consecutive terms.
Harmonic Sequences
Number Pattern Formula for Harmonic Sequences: Tn = 1/ ( a + (n – 1)d ) where n is the ordinal numerical value of the term, a is the denominator of the first term, and d is the common difference between the denominators of any two consecutive terms.
Sequence of Square Numbers
Number Pattern Formula for Square Numbers: Tn = n2 , where n is the ordinal numerical value of the term.
Sequence of Triangular Numbers
Number Pattern Formula for Triangular Numbers: Tn = n(n + 1)/2, where n is the ordinal numerical value of the term.
Fibonacci Sequence
Number Pattern Formula for Fibonacci Sequence: Tn = Tn–1 + Tn–2 , where n is the ordinal numerical value of the term.
Gradient = (y2 – y1)/(x2 – x1) = Change in y/Change in x
Equation of Straight Line
y = mx + c
m = gradient
c = y-intercept (The point when the graph cuts the y-axis)
Coordinate Geometry Formula: Length of Line Segment
Length of AB=√( (y2 – y1)2 + (x2 – x1)2) units
Coordinate Geometry Formula: Equation of Line
For a line with gradient m and passing through the point (x1, y1), the equation of the line is given by: y – y1 = m(x – x1).
**You may still use y = mx + c and substitute (x1, y1) into the equation to find the value of c. But may involve more steps.
Trigonometry
Trigonometric Ratios of Acute Angles
For right‒angled triangle,
1. Sine sin θ = opposite/hypotenuse SOH
2. Cosine cos θ = adjacent/hypotenuse CAH
3. Tangent tanθ = opposite/adjacent TOA
Sine and Cosine of Obtuse Angle
For any acute angle θ,
sin ( 180° – θ) = sin θ
cos ( 180° – θ) = – cos θ
Area of Triangle
For non‒right angled triangle with any 2 given sides and an included angle,
Area of ΔABC = (1/2)absin C
= (1/2)acsin B
= (1/2)bcsin A
Sine rule
For any triangle ABC, a/sin A = b/sin B = c/sin C
where A, B and C are the interior angles
a, b and c are length of their opposite sides respectively.
Cosine rule
For any tiangle ABC,
a2 = b2 + c2 – 2bc cos A cos A is an included angle
b2 = a2 + c2 – 2ac cos B cos B is an included angle
c2 = a2 + b2 – 2ab cos C cos C is an included angle
or
cos A = (b2 + c2 – a2)/2bc
cos B = (a2 + c2 – b2)/2ac
cos C = (a2 + b2 – c2)/2ab
where A, B and C are the interior angles
a, b and c are length of their opposite sides respectively.
Probability
Probability is a measure of chance.
The probability of an event, A is:
P(A) = k/m
Where k is the number of outcomes of A while m is the total number of possible outcomes.
Mutually Exclusive Events and Addition Law
Two events are called mutually exclusive events if they cannot occur at the same time.
Eg. In tossing a coin, the event A of getting 1 head and the event of event B of getting 1 tail are exclusive events. They cannot happen at the same time.
Addition Law
If A and B are exclusive events, then the probability that either A or B occurring is given by
P(A or B) = P(A) + P(B)
If events A and B are not mutually exclusive then P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Independent Events and Multiplication Law
Two events are called independent events if the occurrence of one event does not affect the probability of occurrence of the other event.
Eg. In tossing a coin and a die, the event A that the coin is a head and the event B that the number on the die is even are independent events.
Multiplication Law
If A and B are independent events, then the probability that both A and B occurring is given by
P(A and B) = P(A) × P(B)
Statistics
Standard Deviation
To find the standard deviation of an ungrouped data set {x1, x2, x3, …., xn} , where n is the number of data in the set:
Standard Deviation for Grouped Data
To find the standard deviation of a grouped data set:
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