A-level Mathematics/AQA/MPC2
Transformations of functions edit
Sequences and series edit
Notation edit
— the general term of a sequence; the nth term
— the first term of a sequence
— the last term of a sequence
— the common difference of an arithmetic progression
— the common ratio of a geometric progression
— the sum to n terms:
— the sum of
— infinity (which is a concept, not a number)
— n tends towards infinity (n gets bigger and bigger)
— the modulus of x (the value of x, ignoring any minus signs)
Convergent, divergent and periodic sequences edit
Convergent sequences edit
A sequence is convergent if its nth term gets closer to a finite number, L, as n approaches infinity. L is called the limit of the sequence:
Another way of denoting the same thing is:
Definition of the limit of a convergent sequence edit
Generally, the limit of a sequence defined by is given by
Divergent sequences edit
Sequences that do not tend to a limit as increases are described as divergent. eg: 1, 2, 4, 8, 16, ...
Periodic sequences edit
Sequences that move through a regular cycle (oscillate) are described as periodic.
Series edit
A series is the sum of the terms of a sequence. Those series with a countable number of terms are called finite series and those with an infinite number of terms are called infinite series.
Arithmetic progressions edit
An arithmetic progression, or AP, is a sequence in which the difference between any two consecutive terms is a constant called the common difference. To get from one term to the next, you simply add the common difference:
Expression for the nth term of an AP edit
Formulae for the sum of the first n terms of an AP edit
The sum of an arithmetic progression is called an arithmetic series.
Formulae for the sum of the first n natural numbers edit
The natural numbers are the positive integers, i.e. 1, 2, 3…
Geometric progressions edit
An geometric progression, or GP, is a sequence in which the ratio between any two consecutive terms is a constant called the common ratio. To get from one term to the next, you simply multiply by the common ratio:
Expression for the nth term of an GP edit
Formula for the sum of the first n terms of a GP edit
Formula for the sum to infinity of a GP edit
Binomial theorem edit
The binomial theorem is a formula that provides a quick and effective method for expanding powers of sums, which have the general form .
Binomial coefficients edit
The general expression for the coefficient of the term in the expansion of is:
where
is called n factorial. By definition, .
Binomial expansion of (1+x)n edit
Trigonometry edit
Arc length edit
Sector area edit
Trigonometric identities edit
Indices and logarithms edit
Laws of indices edit
(for x ≠ 0)
Logarithms edit
Laws of logarithms edit
The sum of the logs is the log of the product.
The difference of the logs is the log of the quotient.
The index comes out of the log of the power.
Differentiation edit
Differentiating the sum or difference of two functions edit
Integration edit
Integrating axn edit
Area under a curve edit
The area under the curve between the limits and is given by