IB Mathematics SL/Algebra

Topic 1 - AlgebraEdit

IntroductionEdit

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 SeriesEdit

A series is a sum of numbers. For example,


 


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,


 


Finite and Infinite SequencesEdit

A more formal definition of a finite sequence with terms in a set S is a function from {1,2,...,n} to   for some n ≥ 0.

An infinite sequence in S is a function from {1,2,...} (the set of natural numbers)

ArithmeticEdit

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:

 

Where   is the nth term,   is the first term, d is the difference, and n is the number of terms


Sum of Infinite and Finite Arithmetic SeriesEdit

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:

 .

Geometric Sequences and SeriesEdit

Sum of Finite and Infinite Geometric SeriesEdit

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:

 

 .

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

 

ExponentsEdit

  is the same as  

 

Laws of ExponentsEdit

The algebra section requires an understanding of exponents and manipulating numbers of exponents. An example of an exponential function is   where a is being raised to the   power. An exponent is evaluated by multiplying the lower number by itself the amount of times as the upper number. For example,  . If the exponent is fractional, this implies a root. For example,  . Following are laws of exponents that should be memorized:

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LogarithmsEdit

Laws of LogarithmsEdit

 

 

 

Change of Base formula:

 

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:

 

Binomial TheoremEdit

The Binomial Expansion Theorem is used to expand functions like   without having to go through the tedious work it takes to expand it through normal means

 

For this equation, essentially one would go through the exponents that would occur with the final product of the function ( ). From this   comes in as the coefficient, where   equals the row number of the row from Pascal's Triangle, and   is the specific number from that row.

Ex.  

Pascal's TriangleEdit

                  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