A sequence is an ordered list of numbers; a series is what you get when you add them up. Arithmetic sequences grow by a fixed amount, geometric sequences by a fixed factor.
➕ Arithmetic sequences
Each term is the previous term plus the same common difference d.
Check: subtract any term from the next. If you always get the same number, it is arithmetic.
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An arithmetic sequence adds the same amount each time (d = 3), so it climbs in a straight line. A geometric sequence multiplies by the same ratio (r = 2), so it accelerates.
✖️ Geometric sequences & series
Each term is the previous term times the same common ratio r.
Watch out: an infinite geometric series only has a sum when |r| < 1. If |r| ≥ 1 the terms never shrink, so it diverges.
✏️ Quick check · 1 of 6
Common difference
What is the common difference of 12, 7, 2, −3, …?
Subtract the first term from the second.
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Spot the geometric one
Which sequence is geometric?
Look for a constant ratio, not a constant difference.
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An arithmetic sequence has a₁ = 4 and d = 6.
Find the 20th term.
Enter an exact answer, without rounding.
Use a_n = a₁ + (n − 1)d with n = 20.
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Add the first 15 terms of 2, 5, 8, 11, …
What is S₁₅?
Enter an exact answer, without rounding.
S_n = n/2 · (2a₁ + (n − 1)d).
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A geometric sequence has a₁ = 3 and r = 2.
Find the 8th term.
Enter an exact answer, without rounding.
Use a_n = a₁ · r^(n − 1).
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Consider 12 + 6 + 3 + 1.5 + …
What does this infinite series add up to?
Enter an exact answer, without rounding.
S∞ = a₁ / (1 − r).
📝 Lesson quiz · 5 questions
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Question 1 of 5
An arithmetic sequence has a₁ = 7 and d = −2. Its nth term is:
Question 2 of 5
An infinite geometric series has a finite sum when:
Question 3 of 5
The sum 1 + 2 + 3 + … + n equals:
Question 4 of 5
For 81, 27, 9, 3, … the common ratio is:
Question 5 of 5
In an arithmetic sequence a₃ = 11 and a₇ = 23. The common difference is:
Lesson 2 · Trigonometry
📐 Trig Identities & Equations
Identities are equations that are true for every angle. They let you simplify expressions and turn hard trig equations into ones you can solve.
🔑 The key identities
Everything below follows from the Pythagorean identity and the definition of tangent.
sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
1 + tan²θ = sec²θ
Double angle:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
Example: sin θ = 3/5, θ acute
cos θ = 4/5
sin 2θ = 2 · 3/5 · 4/5 = 24/25
cos 2θ = 1 − 2 · 9/25 = 7/25
🖼️ See it
The same wave shape: cos x is sin x shifted 90° to the left.
🎯 Solving trig equations
Isolate the trig function, find the reference angle, then use the quadrant (ASTC: All, Sin, Tan, Cos positive) to list every solution in the interval.
Solve 2 sin θ − 1 = 0 for 0° ≤ θ < 360°
sin θ = 1/2
reference angle 30°, sine is positive in Q1 and Q2
θ = 30°, 150°
Solve 2cos²θ − cos θ − 1 = 0
(2cos θ + 1)(cos θ − 1) = 0
cos θ = −1/2 or cos θ = 1
θ = 120°, 240° or θ = 0°
Tip: never divide both sides by a trig function: you lose solutions. Factor instead.
✏️ Quick check · 1 of 6
Pythagorean identity
sin²θ + cos²θ equals:
It comes from x² + y² = 1 on the unit circle.
✏️ Quick check · 2 of 6
Double angle
Which expression equals cos 2θ?
There are three equivalent forms of cos 2θ.
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θ is acute and sin θ = 3/5.
Find cos θ as a decimal.
Enter an exact answer, without rounding.
Use sin²θ + cos²θ = 1 and take the positive root.
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θ is acute and sin θ = 3/5.
Find sin 2θ as a decimal.
Enter an exact answer, without rounding.
sin 2θ = 2 sin θ cos θ and cos θ = 0.8.
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Consider 2 sin θ = 1 on 0° ≤ θ < 360°.
How many solutions are there?
Enter an exact answer, without rounding.
Sine is positive in two quadrants.
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θ is acute and sin θ = 3/5.
Find cos 2θ as a decimal.
Enter an exact answer, without rounding.
cos 2θ = 1 − 2sin²θ.
📝 Lesson quiz · 5 questions
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Question 1 of 5
tan θ is equal to:
Question 2 of 5
The solutions of sin θ = 1/2 for 0° ≤ θ < 360° are:
Question 3 of 5
1 + tan²θ equals:
Question 4 of 5
cos(180° − θ) equals:
Question 5 of 5
sin²θ / (1 − cos²θ) simplifies to:
Lesson 3 · Geometry
➡️ Vectors in 2D
A vector has both a size and a direction. Vectors describe velocity, force and displacement, and the dot product tells you how two vectors are aligned.
📏 Components, size and direction
Write a vector as (x, y). Add vectors component by component, and scale them by multiplying every component.
|v| = √(x² + y²)
unit vector v̂ = v / |v|
a = (3, 4), b = (−2, 5)
a + b = (1, 9)
2a − b = (6 + 2, 8 − 5) = (8, 3)
|a| = √(9 + 16) = 5
v̂ for a = (3/5, 4/5)
🖼️ See it
A 2D vector splits into horizontal and vertical components. Its length comes from Pythagoras, and its direction from tan θ = 4/3.
✳️ The dot product
The dot product turns two vectors into one number.
a · b = a₁b₁ + a₂b₂ = |a||b| cos θ
(3, 4) · (−2, 5) = −6 + 20 = 14
Angle: cos θ = a · b / (|a||b|)
(1, 0) · (1, 1) = 1, |a| = 1, |b| = √2
cos θ = 1/√2 → θ = 45°
Perpendicular test: two non-zero vectors are perpendicular exactly when their dot product is 0.
Example: a plane heads north at 200 km/h in a 50 km/h wind blowing east. The resultant velocity is (50, 200), with speed √(50² + 200²) = √42 500 ≈ 206 km/h.
✏️ Quick check · 1 of 6
Magnitude
What is |(6, 8)|?
Use √(x² + y²).
✏️ Quick check · 2 of 6
Perpendicular pair
Which pair of vectors is perpendicular?
Perpendicular vectors have dot product 0.
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a = (3, 4) and b = (−2, 5).
Compute a · b.
Enter an exact answer, without rounding.
Multiply matching components and add.
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a = (3, 4) and b = (−2, 5).
What is the x-component of 2a − b?
Enter an exact answer, without rounding.
Double a, then subtract b component by component.
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A vector is (5, 12).
Find its magnitude.
Enter an exact answer, without rounding.
5-12-13 is a Pythagorean triple.
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The vectors (k, 3) and (6, −2) are perpendicular.
Find k.
Enter an exact answer, without rounding.
Set the dot product equal to 0.
📝 Lesson quiz · 5 questions
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Question 1 of 5
The unit vector in the direction of (3, 4) is:
Question 2 of 5
If a · b = 0 and neither vector is zero, the angle between them is:
Question 3 of 5
(2, −1) + (−5, 4) equals:
Question 4 of 5
The dot product a · b also equals:
Question 5 of 5
What is the magnitude of (−3, 4)?
Lesson 4 · Discrete
🎲 Counting: Permutations & Combinations
Counting without listing: the key question is always whether the order of the chosen items matters.
🧮 Permutations and combinations
The fundamental counting principle: if one choice can be made in m ways and another in n ways, together they can be made in m × n ways. A factorial n! multiplies all whole numbers from n down to 1 (and 0! = 1).
Order matters (permutation): nPr = n! / (n − r)!
Order does not matter (combination): nCr = n! / (r!(n − r)!)
5P3 = 5·4·3 = 60
5C3 = 60 / 3! = 10
Every combination is counted r! times as a permutation.
Ask yourself: if I swap two of the chosen items, is it a different outcome? Yes → permutation. No → combination.
🖼️ See it
Each group of 2 letters can be arranged in 2! = 2 ways, so combinations = permutations ÷ r!.
🔁 Repeats and the binomial theorem
If some items are identical, divide by the factorial of each repeat count.
BANANA: 6 letters, A ×3, N ×2, B ×1
arrangements = 6! / (3! · 2!) = 720 / 12 = 60
The binomial theorem expands (a + b)ⁿ using combinations as coefficients (the rows of Pascal's triangle).
Not binomial: drawing cards without replacement (p changes each draw) or counting trials until the first success (n is not fixed).
✏️ Quick check · 1 of 6
Independent events
Events A and B are independent when:
Think of two separate coin flips.
✏️ Quick check · 2 of 6
Is it binomial?
Which situation is modelled by a binomial distribution?
Fixed n, two outcomes, constant p, independent.
✍️ Type your answer · 3 of 6
P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2.
Find P(A ∪ B).
Enter an exact answer, without rounding.
Subtract the overlap so it is not counted twice.
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X ~ B(20, 0.3).
What is the mean of X?
Enter an exact answer, without rounding.
mean = n × p.
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A fair coin is flipped 5 times.
What is P(exactly 2 heads) as a decimal?
Enter an exact answer, without rounding.
5C2 = 10 and (1/2)⁵ = 1/32.
✍️ Type your answer · 6 of 6
A game pays $10 with probability 0.2 and costs $2 otherwise.
What is the expected gain per play in dollars?
Enter an exact answer, without rounding.
$
Probability-weighted average of the outcomes.
📝 Lesson quiz · 5 questions
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Question 1 of 5
The variance of X ~ B(10, 0.5) is:
Question 2 of 5
A fair coin is flipped 3 times. P(at least one head) is:
Question 3 of 5
If P(A) = 0.3, then P(not A) is:
Question 4 of 5
The probabilities of all outcomes in a sample space add up to:
Question 5 of 5
The expected value of a fair six-sided die roll is:
Lesson 6 · Algebra
🧱 Matrices & Systems of Equations
A matrix is a rectangular grid of numbers. Matrices organise data and give a compact way to solve systems of linear equations.
🔲 Matrix arithmetic
Add or subtract matrices of the same size entry by entry. To multiply, each entry of the product is a row of the first matrix times a column of the second, so an (m × n) matrix times an (n × p) matrix gives an (m × p) matrix.
Shifting vs scaling: adding a constant to every value shifts the mean but not σ. Multiplying every value by k multiplies σ by |k|.
🖼️ See it
About 68% of values lie within one standard deviation of the mean, 95% within two, 99.7% within three.
🔔 The normal distribution
Many measurements (heights, test marks, errors) follow a symmetric bell curve. The empirical rule: about 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ.
z = (x − μ) / σ (how many σ above the mean)
μ = 70, σ = 10
x = 90 → z = 2 (about the top 2.5%)
x = 60 → z = −1 (one σ below the mean)
x = 85 → z = 1.5
✏️ Quick check · 1 of 6
Empirical rule
In a normal distribution, about what percentage of values lie within 1 standard deviation of the mean?
The three numbers are 68, 95 and 99.7.
✏️ Quick check · 2 of 6
Adding a constant
Every value in a data set is increased by 10. What happens?
Does the distance between values change?
✍️ Type your answer · 3 of 6
Data: 2, 4, 4, 4, 5, 5, 7, 9.
Find the mean.
Enter an exact answer, without rounding.
Add all values and divide by 8.
✍️ Type your answer · 4 of 6
Same data: 2, 4, 4, 4, 5, 5, 7, 9 (mean 5).
Find the (population) standard deviation.
Enter an exact answer, without rounding.
Average the squared deviations, then take the square root.
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Scores are normally distributed with mean 70 and σ = 10.
What is the z-score of 85?
Enter an exact answer, without rounding.
z = (x − μ) / σ.
✍️ Type your answer · 6 of 6
Use the empirical rule.
About what percentage of values lie within 2 standard deviations of the mean?
Enter an exact answer, without rounding.
%
It is the middle number of 68–95–99.7.
📝 Lesson quiz · 5 questions
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Question 1 of 5
An extreme outlier added to a data set will usually:
Question 2 of 5
A negative z-score means the value is:
Question 3 of 5
Which measure of centre is least affected by outliers?
Question 4 of 5
If σ = 5, the variance is:
Question 5 of 5
The normal curve is symmetric about:
Lesson 8 · Calculus
📉 Introduction to Derivatives
A derivative measures how fast something changes at one instant: the slope of a curve at a point. It is the central idea of calculus.
🏃 Rates of change and the derivative
The average rate of change of f between x = a and x = b is the slope of the secant line:
average rate = (f(b) − f(a)) / (b − a)
f(x) = x² between x = 1 and x = 3:
(9 − 1) / (3 − 1) = 4
Let the two points get closer and closer together. The slope settles on the derivative f′(x): the instantaneous rate of change, or the slope of the tangent line.
🖼️ See it
At x = 2 the derivative 2x equals 4, the slope of the tangent line.