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

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. $7_{5}=35$

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