# 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,

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+...}$

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,

${\displaystyle 1,2,3,4,5,...}$

### 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 ${\displaystyle S}$  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.

    1, 2, 3, 4, 5, ...


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:

${\displaystyle \ u_{n}=u_{1}+(n-1)d.}$

Where ${\displaystyle u_{n}}$  is the nth term, ${\displaystyle u_{1}}$  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:

${\displaystyle S_{n}={\frac {n}{2}}\cdot (2u_{1}+(n-1)d)={\frac {n}{2}}\cdot (u_{1}+u_{n})}$ .

## 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:

{\displaystyle {\begin{aligned}u_{n}=u_{1}\cdot r^{n-1}&.\end{aligned}}}

${\displaystyle S_{n}={\frac {u_{1}(r^{n}-1)}{r-1}}={\frac {u_{1}(1-r^{n})}{1-r}}}$ .

The sum of all terms (an infinite geometric sequence): If -1 < r < 1, then

${\displaystyle S={\frac {u_{1}}{1-r}}}$

## Exponents

${\displaystyle a^{x}=b}$  is the same as ${\displaystyle log_{a}\cdot b=x}$

{\displaystyle {\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 ${\displaystyle a^{c}}$  where a is being raised to the ${\displaystyle c^{th}}$  power. An exponent is evaluated by multiplying the lower number by itself the amount of times as the upper number. For example, ${\displaystyle 2^{3}=2\times 2\times 2=8}$ . If the exponent is fractional, this implies a root. For example, ${\displaystyle 4^{\frac {1}{2}}={\sqrt {4}}=2}$ . Following are laws of exponents that should be memorized:

• ${\displaystyle a^{m}a^{n}=a^{m+n}}$
• ${\displaystyle (ab)^{m}=a^{m}b^{m}}$
• ${\displaystyle (a^{m})^{n}=a^{mn}}$
• ${\displaystyle a^{m/n}={\sqrt[{n}]{a^{m}}}}$

## Logarithms

### Laws of Logarithms

${\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y\,\!}$

${\displaystyle \log _{b}({\frac {x}{y}})=\log _{b}x-\log _{b}y}$

${\displaystyle \log _{b}x^{y}=y\log _{b}x\,\!}$

Change of Base formula:

${\displaystyle \log _{b}(a)={\frac {\log _{c}(a)}{\log _{c}(b)}}.}$

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:

${\displaystyle \log _{2}(16)={\frac {\log(16)}{\log(2)}}.}$

## Binomial Theorem

The Binomial Expansion Theorem is used to expand functions like ${\displaystyle (x+y)^{n}}$  without having to go through the tedious work it takes to expand it through normal means

${\displaystyle (x+y)^{n}=_{n}C_{0}x^{n}y^{0}+_{n}C_{1}x^{n-1}y^{1}+_{n}C_{2}x^{n-2}y^{2}+...+_{n}C_{r}x^{n-r}y^{r}+...+_{n}C_{n}x^{0}y^{n}\,\!}$

For this equation, essentially one would go through the exponents that would occur with the final product of the function (${\displaystyle x^{n}y^{0}+x^{n-1}y^{1}+x^{n-2}y^{2}+...+x^{0}y^{n}}$ ). From this ${\displaystyle C_{n}}$  comes in as the coefficient, where ${\displaystyle C}$  equals the row number of the row from Pascal's Triangle, and ${\displaystyle n}$  is the specific number from that row.

Ex. ${\displaystyle 7_{5}=35}$

### Pascal's Triangle

                  1                      =Row 0
1   1                    =Row 1
1   2   1                  =Row 2
1   3   3   1                =Row 3
1   4   6   4   1              =Row 4
1   5  10  10   5   1            =Row 5
1   6  15  20  15   6   1          =Row 6
1   7  21  35  35  21   7   1        =Row 7
1   8  28  56  70  56  28   8   1      =Row 8
1   9   36 84 126 126  84  36   9   1    =Row 9