A-level Mathematics/AQA/MPC2

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Transformations of functions[edit | edit source]

Sequences and series[edit | edit source]

Notation[edit | edit source]

— 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 | edit source]

Convergent sequences[edit | edit source]

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 | edit source]

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

Divergent sequences[edit | edit source]

Sequences that do not tend to a limit as increases are described as divergent. eg: 1, 2, 4, 8, 16, ...

Periodic sequences[edit | edit source]

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

Series[edit | edit source]

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 | edit source]

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 | edit source]

Formulae for the sum of the first n terms of an AP[edit | edit source]

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

Formulae for the sum of the first n natural numbers[edit | edit source]

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

Geometric progressions[edit | edit source]

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 | edit source]

Formula for the sum of the first n terms of a GP[edit | edit source]

Formula for the sum to infinity of a GP[edit | edit source]

Binomial theorem[edit | edit source]

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 | edit source]

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 | edit source]

Trigonometry[edit | edit source]

Arc length[edit | edit source]

Sector area[edit | edit source]

Trigonometric identities[edit | edit source]

Indices and logarithms[edit | edit source]

Laws of indices[edit | edit source]

(for x ≠ 0)

Logarithms[edit | edit source]

Laws of logarithms[edit | edit source]

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 | edit source]

Differentiating the sum or difference of two functions[edit | edit source]

Integration[edit | edit source]

Integrating axn[edit | edit source]

Area under a curve[edit | edit source]

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