$$u_n=u_1+(n-1)d,\quad S_n=\frac{n}{2}(u_1+u_n)=\frac{n}{2}[2u_1+(n-1)d].$$

$$u_n=u_1\,r^{n-1},\quad S_n=\frac{u_1(r^n-1)}{r-1}\;(r\neq1),\quad S_\infty=\frac{u_1}{1-r}\;(|r|<1).$$

$$(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r,\quad\binom{n}{r}=\frac{n!}{r!(n-r)!}.$$

Direct proof starts from a definition or an already-established result and reasons forward, step by justified step, to the claim. A standard opening move: write an even integer as $2k$ and an odd integer as $2k+1$ for some integer $k$, then manipulate algebraically.

Proof by exhaustion is only valid when the claim can be split into a *finite* number of cases that together cover every possibility (e.g. checking all residues mod $n$) — each case is then verified separately.

Disproof by counterexample is the standard way to show a universal statement ("for all $n$, …") is false: exhibiting one value of $n$ for which it fails is a complete, rigorous disproof — no further argument is needed, and no number of confirming cases can substitute for it if a single counterexample exists.

$$\text{Compound interest: }FV=PV\!\left(1+\frac{r}{100k}\right)^{kn}.$$

Use GDC TVM solver: $N$ = total periods, $I\%$, $PV$, $PMT$, $FV$, $P/Y=C/Y$.

Domain: set of permitted inputs. Range: set of possible outputs. Composite: $(f\circ g)(x)=f(g(x)).$ Inverse: $f^{-1}$ exists iff $f$ is one-to-one; $f(f^{-1}(x))=x.$

Graphically: $f^{-1}$ is the reflection of $f$ over $y=x.$

$$\text{Circumference }=2\pi r,\quad\text{Area }=\pi r^2,\quad\text{Arc length }=r\theta,\quad\text{Sector area }=\tfrac{1}{2}r^2\theta.$$

One radian is the angle subtended at a circle's centre by an arc equal in length to the radius. Since the full circumference $2\pi r$ wraps around once, a full turn is $2\pi$ radians — the same $360°$ as always, just measured differently: $$2\pi\text{ rad}=360°\quad\Longrightarrow\quad \theta_{\text{rad}}=\theta_{\text{deg}}\times\frac{\pi}{180},\qquad\theta_{\text{deg}}=\theta_{\text{rad}}\times\frac{180}{\pi}.$$

Radians are the natural unit for the circle formulae above: arc length $=r\theta$ and sector area $=\tfrac12r^2\theta$ are only correct when $\theta$ is in radians (in degrees, arc length is $r\theta\times\frac{\pi}{180}$ instead) — exactly because "radian" is defined so that arc length divided by radius is the angle, with no conversion factor needed.

$$\sin^2\theta+\cos^2\theta=1,\quad\tan\theta=\frac{\sin\theta}{\cos\theta},$$$$\sin2\theta=2\sin\theta\cos\theta,\quad\cos2\theta=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta.$$

$$\mathbf{v}=\begin{pmatrix}v_1\\v_2\end{pmatrix},\quad|\mathbf{v}|=\sqrt{v_1^2+v_2^2},\quad\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2=|\mathbf{u}||\mathbf{v}|\cos\theta.$$

Vector equation of a line: $\mathbf{r}=\mathbf{a}+t\mathbf{b}.$

$$\bar{x}=\frac{\sum f_ix_i}{\sum f_i}\quad(\text{mean from frequency table}),\quad IQR=Q_3-Q_1.$$

A value is an outlier if $x<Q_1-1.5\,IQR$ or $x>Q_3+1.5\,IQR.$

A permutation counts arrangements where order matters; a combination counts selections where it doesn't. Choosing $r$ objects from $n$ distinct objects: $${}^nP_r=\frac{n!}{(n-r)!}\quad(\text{ordered}),\qquad{}^nC_r=\binom{n}{r}=\frac{n!}{r!(n-r)!}\quad(\text{unordered}).$$ ${}^nC_r$ is exactly the binomial coefficient from Chapter 1's binomial theorem — the number of ways to choose which $r$ of the $n$ factors contribute the $b$ in each term of $(a+b)^n$ is what makes $\binom{n}{r}$ show up there, and the same counting question — how many size-$r$ subsets, or how many size-$k$ "successes" among $n$ trials — is exactly what makes $\binom{n}{k}$ appear again below in the binomial distribution.

$$\chi^2_{\text{calc}}=\sum\frac{(f_o-f_e)^2}{f_e},\quad f_e=\frac{\text{row total}\times\text{col total}}{\text{grand total}}.$$

Reject $H_0$ (independence) if $\chi^2_{\text{calc}}>\chi^2_{\text{crit}}$ or $p<$ significance level.

$$\frac{d}{dx}(x^n)=nx^{n-1},\quad\frac{d}{dx}(e^x)=e^x,\quad\frac{d}{dx}(\ln x)=\frac{1}{x},$$$$\frac{d}{dx}(\sin x)=\cos x,\quad\frac{d}{dx}(\cos x)=-\sin x,\quad\frac{d}{dx}(\tan x)=\sec^2 x.$$

$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\;(n\neq-1),\quad\int e^x\,dx=e^x+C,\quad\int\frac{1}{x}\,dx=\ln|x|+C.$$$$\int\sin x\,dx=-\cos x+C,\quad\int\cos x\,dx=\sin x+C.$$$$\int_a^b f(x)\,dx=F(b)-F(a)\quad(\text{Fundamental Theorem of Calculus}).$$

Area between curves: $\displaystyle\int_a^b[f(x)-g(x)]\,dx$ where $f\ge g$ on $[a,b].$