Imaginary Numbers

$i=\sqrt{-1}$

Imaginary numbers can be graphed in the cartesian plane

These coordinates are not expressed as ordered pairs, rather as sums (e.g. $(2+3i)$)

$i^2=-1, i^3=-i, i^4=1$. This pattern continues to infinity.

Sequences and Series

Sequence: an ordered list

Series: the sum of an ordered list

Two types: arithmetic (adding/subtracting, “common difference”) and geometric (multiplying/dividing, “common ratio”)

Can be written in sigma notation:$\displaystyle\sum_{i=a}^b\{pattern\}$

The capital greek letter Sigma (the big thing that looks like an E) means “the sum of”

The number on the bottom is the index, or the first number in the series

The number on top is the number of terms summed

The pattern is the pattern which the numbers to be plugged into the common difference/ratio, which can be determined by looking at the first few terms of the progression

Arithmetic series formulas:

To find a term: $a_n=a_1+(n-1)d$ ($d$ is the common difference)

To find the sum: $S_n=\frac{a_1+a_n}{2}n$ or $S_n=\frac{n}{2}[2a_1+(n-1)d]$

Geometric series formulas:

To find a term: $a_n=a_1\cdot r^{n-1}$ ($r$ is the common ratio)

To find a sum: $S_n=\frac{a_1(1-r^n)}{1-r}$

Pascal’s triangle

Useful for binomial expansion, amongst other things; terms represent coefficients of binomial expansions of $(x+1)^n$

A “triangle” of numbers which follows the following pattern: add the numbers directly adjacent to a value to that value and write them diagonally down. Blank spaces count as zeros.

For any given row, that is how the pattern for the polynomial expansion will look (e.g. the third row will be the polynomial expansion of $x^3$)

Other Review

$\sin^2\theta$ and other squared circular functions look and behave exactly the same as they normally would except they do not dip below the $x$ axis

$n!$ means $n$ factorial or all the numbers before $n$ except for 0 multiplied by each other. Note: 0!=1. This is because the definition of a factorial is the number of ways you can arrange $n$ things

Fibonacci numbers: all numbers following the fibonacci sequence. That is, $F_n = F_{n-1} + F_{n-2}$. Or, the next number in the set is obtained by adding the two previous numbers. The first few Fibonacci numbers are 1,1,2,3,5,8,13,21...

Triangular numbers: number of units/points/things needed to form a triangle. The first few are 1,3,6,10,15,21