# 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.

    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:

$\ u_n = u_1 + (n - 1)d.$

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

The nth term of a geometric sequence:

\begin{align} u_n = u_1\cdot r^{n - 1}&. \end{align}

$S_n=\frac{u_1( r^n-1)}{r-1}=\frac{u_1( 1-r^n)}{1-r}$.

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{align} a^x = e^{x \cdot\ln a}\, \end{align}

### 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:

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

## Logarithms

### Laws of Logarithms

$\log_b(xy) = \log_bx + \log_by \,\!$

$\log_b(\frac{x}{y}) = \log_bx - \log_by$

$\log_bx^y = y\log_bx \,\!$

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

$(x+y)^n = _nC_0x^ny^0+_nC_1x^{n-1}y^1 + _nC_2x^{n-2}y^2 + ... +_nC_rx^{n-r}y^r+... + _nC_nx^0y^n\,\!$

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

Ex. $7_5=35$

### 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