Core 20
Questions

Geometric Sequence: \(u_n = u_1 \cdot r^{n-1}\), Sum: \(S_n = \frac{u_1(1-r^n)}{1-r}\)

Arithmetic Sequence: \(u_n = u_1 + (n-1)d\), Sum: \(S_n = \frac{n}{2}(2u_1+(n-1)d)\)

Binomial Theorem: \((a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\)

Infinite Geo Sum (|r|<1): \(S_\infty = \frac{u_1}{1-r}\)

Laws of Logarithms: \(\log(ab)=\log a+\log b\), \(\log\!\left(\frac{a}{b}\right)=\log a-\log b\), \(\log(a^n)=n\log a\)

Quadratic: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), discriminant \(\Delta = b^2-4ac\)

Vertex form: \(f(x)=a(x-h)^2+k\), vertex at \((h,k)\)

Exponential: \(f(x)=a\cdot b^x\), growth if \(b>1\), decay if \(0<b<1\)

Inverse function: swap \(x\) and \(y\), solve for \(y\). Domain/range swap.

Transformation: \(f(x-a)+b\) shifts right \(a\), up \(b\)

Sine Rule: \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)

Cosine Rule: \(a^2=b^2+c^2-2bc\cos A\)

Area of triangle: \(\frac{1}{2}ab\sin C\)

Radians: \(\pi \text{ rad} = 180°\). Arc length \(s=r\theta\), Area sector \(A=\frac{1}{2}r^2\theta\)

Unit circle: \(\sin^2\theta+\cos^2\theta=1\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}\)

Combined events: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)

Conditional prob: \(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\)

Independent events: \(P(A\cap B)=P(A)\cdot P(B)\)

Binomial distribution: \(X\sim B(n,p)\), \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\)

Normal distribution: \(X\sim N(\mu,\sigma^2)\), standardise: \(z=\frac{x-\mu}{\sigma}\)

Derivative rules: \(\frac{d}{dx}[x^n]=nx^{n-1}\), \(\frac{d}{dx}[e^x]=e^x\), \(\frac{d}{dx}[\ln x]=\frac{1}{x}\)

Chain rule: \(\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\)

Product rule: \((uv)'=u'v+uv'\)

Integration: \(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\), \(\int e^x\,dx=e^x+C\)

Definite integral = area under curve. \(\int_a^b f(x)\,dx\)

Stationary points: set \(f'(x)=0\), classify with \(f''(x)\)