IB Mathematics SL/Algebra
Topic 1 - Algebra edit
Introduction edit
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 edit
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 Sequences edit
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)
Arithmetic edit
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 Series edit
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 Series edit
Sum of Finite and Infinite Geometric Series edit
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
Exponents edit
is the same as
Laws of Exponents edit
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:
Logarithms edit
Laws of Logarithms edit
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 Theorem edit
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 Triangle edit
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