A-level Mathematics/AQA/MPC2

Transformations of functionsEdit

Sequences and seriesEdit

NotationEdit

  — 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 sequencesEdit

Convergent sequencesEdit

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 sequenceEdit

Generally, the limit   of a sequence defined by   is given by  

Divergent sequencesEdit

Sequences that do not tend to a limit as   increases are described as divergent. eg: 1, -1 , 1 -1

Periodic sequencesEdit

Sequences that move through a regular cycle (oscillate) are described as periodic.

SeriesEdit

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 progressionsEdit

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 APEdit

 

Formulae for the sum of the first n terms of an APEdit

The sum of an arithmetic progression is called an arithmetic series.

 

 

Formulae for the sum of the first n natural numbersEdit

The natural numbers are the positive integers, i.e. 1, 2, 3…

 

Geometric progressionsEdit

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 GPEdit

 

Formula for the sum of the first n terms of a GPEdit

 

 

Formula for the sum to infinity of a GPEdit

 

Binomial theoremEdit

The binomial theorem is a formula that provides a quick and effective method for expanding powers of sums, which have the general form  .

Binomial coefficientsEdit

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)nEdit

 

 

 

TrigonometryEdit

Arc lengthEdit

 

Sector areaEdit

 

Trigonometric identitiesEdit

 

 

Indices and logarithmsEdit

Laws of indicesEdit

 

 

 

  (for x ≠ 0)

 

 

 

LogarithmsEdit

 

 

 

 

Laws of logarithmsEdit

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.

 

DifferentiationEdit

Differentiating the sum or difference of two functionsEdit

 

IntegrationEdit

Integrating axnEdit

 

Area under a curveEdit

The area under the curve   between the limits   and   is given by