A-level Mathematics/AQA/MPC2

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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, -1 , 1 -1

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