User:Wylve/mathsl

Topic 1: Algebra

This topic introduces candidates to basic algebraic concepts and applications. Number systems are now in the presumed knowledge section.

Sequences and Series

Sequences

A sequence is a ordered list of consecutive terms. Below is an example of a sequence:

$2,4,6,8,10,...$

Each number is called a term (separated by a comma) and a sequence always starts with its first term, denoted as $u_{1}$. In the example above, 2 is the first term, and 4 is the second term. A sequence does not have an ending term and can go on forever.

The IB mathematics standard level course explores two types of number sequences: arithmetic sequence and geometric sequence.

Arithmetic Sequences

The arithmetic sequence is a sequence that has consecutive terms increasing by a constant, or the common difference. The common difference is denoted as $d$.

$18,14,10,6,2,-\!2,-\!6,...$

The example above is an arithmetic sequence, with a common difference of -4, as the terms increase by -4, or in other words, decrease by 4. The common difference does not change throughout the sequence.

The nth term of an arithmetic sequence can be found using the general formula:

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

Where $u_{n}$ is the nth term, $u_{1}$ is the first term, d is the difference, and n is the number of terms

A series is a sum of numbers. For example,

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

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)

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 sequence 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:

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

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

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 $log_{a}\cdot b$

${\begin{aligned}a^{x}=e^{x\cdot \ln a}\,\end{aligned}}$

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

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

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 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