A-level Mathematics/AQA/MPC2
Transformations of functions
editSequences and series
editNotation
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
editConvergent sequences
editA 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
editGenerally, the limit of a sequence defined by is given by
Divergent sequences
editSequences that do not tend to a limit as increases are described as divergent. eg: 1, 2, 4, 8, 16, ...
Periodic sequences
editSequences that move through a regular cycle (oscillate) are described as periodic.
Series
editA 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
editAn 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
editThe sum of an arithmetic progression is called an arithmetic series.
Formulae for the sum of the first n natural numbers
editThe natural numbers are the positive integers, i.e. 1, 2, 3…
Geometric progressions
editAn 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
editThe binomial theorem is a formula that provides a quick and effective method for expanding powers of sums, which have the general form .
Binomial coefficients
editThe 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
editArc length
edit
Sector area
edit
Trigonometric identities
edit
Indices and logarithms
editLaws of indices
edit
(for x ≠ 0)
Logarithms
edit
Laws of logarithms
editThe 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
editDifferentiating the sum or difference of two functions
edit
Integration
editIntegrating axn
edit
Area under a curve
editThe area under the curve between the limits and is given by