IB Mathematics (HL)/Algebra

Topic 1: Core - Algebra

Sequences and Series

Arithmetic Sequences and Series

An arithmetic sequence is a sequence that fits the form:

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

where $u_{n}$ is the $n^{th}$ term of the sequence, $u_{1}$ is the first term of the series, and d is called the common difference. The unique characteristic of this sequence is that the next term of the sequence is the previous term added to the common difference. An example of such a sequence is:

$1,2,3,4,...$

where the common difference is 1 and the first term is 1. The general form for an arithmetic sequence that can be made using the formula above is:

$u_{1},u_{1}+d,u_{1}+2d,u_{1}+3d,...$

An arithmetic series is the sum of all the parts of an arithmetic sequence. Using the example above, a series can be written as:

$1+2+3+4+...$

The sum of a finite arithmetic series to the $n^{th}$ term "$u_{n}$" is represented in the equation:

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

The sum can also be written in sigma notation:

$\sum _{n=1}^{k}\left[u_{1}+d(n-1)\right]$

where $u_{k}=u_{n}$. All arithmetic series diverge as the number of terms approaches infinity. This is why the sum of any arithmetic series can only be found to a finite number of terms.

Geometric Sequences and Series

A geometric sequence is a sequence that fits the form:

$u_{n}=u_{1}r^{n-1}$

where $u_{n}$ is the $n^{th}$ term of the sequence, $u_{1}$ is the first term of the series, and r is called the common ratio. In such a sequence, the next term is obtained by multiplying the previous term with the common ratio. An example of a geometric sequence is:

$1,3,9,27...$

A geometric series is the sum of all the parts of a geometric sequence. For the above example, the geometric sequence is:

$1+3+9+27...$

The sum of a finite geometric sequence to the $n^{th}$ term $u_{n}$ is given by the expression:

$S_{n}={\frac {u_{1}(r^{n}-1)}{r-1}}={\frac {u_{1}(1-r^{n})}{1-r}},r\neq 1$

The sum can also be written using sigma notation:

$\sum _{n=1}^{k}u_{1}r^{n-1}$

where $u_{k}=u_{n}$. Unlike arithmetic series, geometric series do not necessarily diverge. If the common ratio, r, is such that $\left|r\right|>1$, then the series diverges. If $\left|r\right|<1$, then the series converges, and as n approaches infinity, the value of the $n^{th}$ term of the sequence approaches zero.

In the case of a diverging geometric series, the sum can only be calculated for finite values of n. However, in the case of a converging geometric series, the sum of an infinite number of terms is given by the expression: $S_{\infty }={\frac {a}{1-r}}$ OR $\lim _{n\to \infty }S_{n}={\frac {a}{1-r}}$

Compound Interest and Population Growth

Based off the idea of arithmetic series is that of compound interest. Consider a case where $P is invested at a rate of r and compunded n times per year. Time is in years and is denoted t. This will be the progression:

$P_{0}=P\,$

$P_{1}=P+{\begin{matrix}{\frac {r}{n}}\end{matrix}}P=P(1+{\begin{matrix}{\frac {r}{n}}\end{matrix}})$

$P_{2}=P(1+{\begin{matrix}{\frac {r}{n}}\end{matrix}})+({\begin{matrix}{\frac {r}{n}}\end{matrix}})(P(1+{\begin{matrix}{\frac {r}{n}}\end{matrix}}))=P(1+{\begin{matrix}{\frac {r}{n}}\end{matrix}})^{2}$

$P_{n}=P(1+{\begin{matrix}{\frac {r}{n}}\end{matrix}})^{nt}$

The same function holds true for population growth where P is the initial number of people, and r is the rate that the population increases.

Exponents and Logarithms

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:

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

Logarithms

It also requires understanding of logarithms. Logs can be used to sove exponential equations such as $2^{x}=8$ by either logging both sides or applying log rules. The default base number on a log is 10. The log base represents, in exponent form, the number that is being raised to a power. The number that is being logged is the exponent. By relating logs to exponents, we can devise that $2^{x}=8$ is the same as $\log _{2}8=x$. Log rules are as follows:

• $\log _{a}a=1$
• $\log _{b}a^{x}=x\log _{b}a$
• $\log _{a}b=b^{a}$
• $\log _{b}a={\frac {\log _{c}a}{\log _{c}b}}$

The same rules apply for $\ln$ as it is the same as $\log _{e}$

Combinations and Binomial Theorem

This theorem is related to Binomials.

Complex Numbers

Complex numbers are numbers that include the square root of a negative number. The square root of -1 is i.

De Moivre's Theorem

De Moivre's Theorem is the expansion of the polar form of co-ordinates on an argand diagram. The form is expressed below.

$\left(\cos x+i\sin x\right)^{n}=\cos \left(nx\right)+i\sin \left(nx\right).\,$


Created by: JTLenzz and Lance

Edits made by: lichking21st on 28/03/2013