# Topic 1 - Algebra
## Introduction

The aim of this section is to introduce candidates to some basic algebraic concepts and applications. Number systems are now in the presumed knowledge section.

## Sequences and Series

A series is a sum of numbers. For example,
$1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+...$
A sequence is a list of numbers, usually separated by a comma. The order in which the numbers are listed is important, so for instance,
$1,2,3,4,5,...$

### Finite and Infinite Sequences

A more formal definition of a finite sequence with terms in a set S is a function from {1,2,...,n} to $S$ for some n ≥ 0.
An infinite sequence in S is a function from {1,2,...} (the set of natural numbers)

### Arithmetic

Arithmetic series or sequences simply involve addition.
Is an example of addition, where 1 is added each time to the prior term.
The formula for finding the nth term of an arithmetic sequence is:
    $\ u_{n}=u_{1}+(n-1)d.$
Where $u_{n}$ is the nth term, $u_{1}$ is the first term, d is the difference, and n is the number of terms

#### Sum of Infinite and Finite Arithmetic Series

An infinite arithmetic series is an infinite series whose terms are in an arithmetic progression. Examples are 1 + 1 + 1 + 1 + · · · and 1 + 2 + 3 + 4 + · · ·.
The sum (Sn) of a finite series is:
$S_{n}={\frac {n}{2}}\cdot (2u_{1}+(n-1)d)={\frac {n}{2}}\cdot (u_{1}+u_{n})$.

## Geometric Sequences and Series
### Sum of Finite and Infinite Geometric Series

A geometric series is a series with a constant ratio between successive terms. Each successive term can be obtained by multiplying the previous term by 'r'
The nth term of a geometric sequence:
    ${\begin{aligned}u_{n}=u_{1}\cdot r^{n-1}&.\end{aligned}}$
$S_{n}={\frac {u_{1}(r^{n}-1)}{r-1}}={\frac {u_{1}(1-r^{n})}{1-r}}$.
The sum of all terms (an infinite geometric sequence): If -1 < r < 1, then
$S={\frac {u_{1}}{1-r}}$

## Exponents

$a^{x}=b$ is the same as $log_{a}\cdot b=x$
${\begin{aligned}a^{x}=e^{x\cdot \ln a}\,\end{aligned}}$

### Laws of Exponents

The algebra section requires an understanding of exponents and manipulating numbers of exponents. An example of an exponential function is $a^{c}$ where a is being raised to the $c^{th}$ power. An exponent is evaluated by multiplying the lower number by itself the amount of times as the upper number. For example, $2^{3}=2\times 2\times 2=8$. If the exponent is fractional, this implies a root. For example, $4^{\frac {1}{2}}={\sqrt {4}}=2$. Following are laws of exponents that should be memorized:
* $a^{m}a^{n}=a^{m+n}$
* $(ab)^{m}=a^{m}b^{m}$
* $(a^{m})^{n}=a^{mn}$
* $a^{m/n}={\sqrt[{n}]{a^{m}}}$

## Logarithms
### Laws of Logarithms

$\log _{b}(xy)=\log _{b}x+\log _{b}y\,\!$
$\log _{b}({\frac {x}{y}})=\log _{b}x-\log _{b}y$
$\log _{b}x^{y}=y\log _{b}x\,\!$
Change of Base formula:
    $\log _{b}(a)={\frac {\log _{c}(a)}{\log _{c}(b)}}.$
Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with a calculator:
    $\log _{2}(16)={\frac {\log(16)}{\log(2)}}.$

## Binomial Theorem

The Binomial Expansion Theorem is used to expand functions like $(x+y)^{n}$ without having to go through the tedious work it takes to expand it through normal means
$(x+y)^{n}=_{n}C_{0}x^{n}y^{0}+_{n}C_{1}x^{n-1}y^{1}+_{n}C_{2}x^{n-2}y^{2}+...+_{n}C_{r}x^{n-r}y^{r}+...+_{n}C_{n}x^{0}y^{n}\,\!$
For this equation, essentially one would go through the exponents that would occur with the final product of the function ($x^{n}y^{0}+x^{n-1}y^{1}+x^{n-2}y^{2}+...+x^{0}y^{n}$). From this $C_{n}$ comes in as the coefficient, where $C$ equals the row number of the row from Pascal's Triangle, and $n$ is the specific number from that row.
Ex. $7_{5}=35$