Concepts

arithmetic sequence

exponential sequence

exponential laws

sigma notation

log laws

the binomial theorm and its extended form

partial fraction

counting principles (combination and permutation)

proof by induction/deduction/contradiction

complex number laws and nature

system of linear equation

Proof

Induction: 1. when \(n=1\), LHS = … ; RHS = … ; LHS = RHS ; the statement is true for n = 1 2. Assume the statement is true for k = n, 3. when n = k+1 4. since true for n=1, and true for n=k+1 whenever n is true, so [Original Statement] is true for all n

contradiction 1. there exists … (a statement to be proven false, often the negate of the statement to be proven true) 2. some deduction 3. XXX gives contradiction 4. done

if contradict integer: even/odd contradiction

敌人图鉴

sequence

大部分题，把条件用表示第n项或和的公式表达出来，解方程即可。

binomial theorem

n项的多项式展开（\((a+b+c)^6\))，通过合并其中几项转化为二项式: \(((a+b)+c)^6\)

complex number

solve complex number equation, make \(z\) into form of \(x+yi\) and 系数对应

\(z^n=\text{some complex number}\)的方程： - RHS into form of: \(r \times cis{(\theta+2\pi k)}, \\ k \in \mathbb{Z}, \\ -\pi<\theta<\pi\), since \(\theta\) is recurrent (==REMEMBER \(\theta\)’s domain!!!==) - then from Demore theorm, \(z=r^{\frac{1}{n}}\times cis(\frac{\theta+2\pi k}{n})\) - feature: there are \(n\) solutions; evenly spaced on a circle with radius of \(r^{\frac{1}{n}}\) center at origin.

when in doubt, draw vector diagram on Argand - then find 菱形 or solve triangle with cos rule

when finding \(\arg{(z)}\), pay attention to the domain