# A-level Mathematics/AQA/MPC2

## Sequences and series

### Notation

${\displaystyle u_{n}\,\!}$ — the general term of a sequence; the nth term

${\displaystyle a\,\!}$ — the first term of a sequence

${\displaystyle l\,\!}$ — the last term of a sequence

${\displaystyle d\,\!}$ — the common difference of an arithmetic progression

${\displaystyle r\,\!}$ — the common ratio of a geometric progression

${\displaystyle S_{n}\,\!}$ — the sum to n terms: ${\displaystyle S_{n}=u_{1}+u_{2}+u_{3}+\ldots +u_{n}\,\!}$

${\displaystyle \sum \,\!}$ — the sum of

${\displaystyle \infty \,\!}$ — infinity (which is a concept, not a number)

${\displaystyle n\rightarrow \infty \,\!}$n tends towards infinity (n gets bigger and bigger)

${\displaystyle |x|\,\!}$ — the modulus of x (the value of x, ignoring any minus signs)

### Convergent, divergent and periodic sequences

#### Convergent sequences

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:

${\displaystyle {\mbox{As }}n\to \infty {\mbox{, }}u_{n}\to L\,\!}$

Another way of denoting the same thing is:

${\displaystyle \lim _{n\to \infty }u_{n}=L\,\!}$

#### Definition of the limit of a convergent sequence

Generally, the limit ${\displaystyle L\,\!}$ of a sequence defined by ${\displaystyle u_{n+1}=f(u_{n})\,\!}$ is given by ${\displaystyle L=f(L)\,\!}$

#### Divergent sequences

Sequences that do not tend to a limit as ${\displaystyle n}$ increases are described as divergent. eg: 1, -1 , 1 -1

#### Periodic sequences

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

### Series

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

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:

${\displaystyle u_{n+1}=u_{n}+d\,\!}$

#### Expression for the nth term of an AP

${\displaystyle u_{n}=a+(n-1)d\,\!}$

#### Formulae for the sum of the first n terms of an AP

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

${\displaystyle S_{n}={\frac {n}{2}}\left\lbrack 2a+(n-1)d\right\rbrack \,\!}$

${\displaystyle S_{n}={\frac {n}{2}}(a+l)\,\!}$

#### Formulae for the sum of the first n natural numbers

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

${\displaystyle S_{n}={\frac {n}{2}}(n+1)\,\!}$

### Geometric progressions

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:

${\displaystyle u_{n+1}=ru_{n}\,\!}$

#### Expression for the nth term of an GP

${\displaystyle u_{n}=ar^{n-1}\,\!}$

#### Formula for the sum of the first n terms of a GP

${\displaystyle S_{n}=a\left({\frac {1-r^{n}}{1-r}}\right)\,\!}$

${\displaystyle S_{n}=a\left({\frac {r^{n}-1}{r-1}}\right)\,\!}$

#### Formula for the sum to infinity of a GP

${\displaystyle S_{\infty }=\sum _{n=1}^{\infty }ar^{n-1}={\frac {a}{1-r}}\qquad {\mbox{where }}-1

### Binomial theorem

The binomial theorem is a formula that provides a quick and effective method for expanding powers of sums, which have the general form ${\displaystyle (a+b)^{n}}$.

#### Binomial coefficients

The general expression for the coefficient of the ${\displaystyle (r+1)^{th}}$ term in the expansion of ${\displaystyle (1+x)^{n}}$ is:

${\displaystyle {}^{n}\!C_{r}={\binom {n}{r}}={\frac {n!}{r!(n-r)!}}}$

where ${\displaystyle n!=1\times 2\times 3\times \ldots \times n}$

${\displaystyle n!}$ is called n factorial. By definition, ${\displaystyle 0!=1}$.

#### Binomial expansion of (1+x)n

${\displaystyle (1+x)^{n}=1+{\binom {n}{1}}x+{\binom {n}{2}}x^{2}+{\binom {n}{3}}x^{3}+\ldots +x^{n}}$

${\displaystyle (1+x)^{n}=1+nx+{\frac {n(n-1)}{2!}}+{\frac {n(n-1)(n-2)}{3!}}+\ldots +x^{n}}$

${\displaystyle (1+x)^{n}=\sum _{r=0}^{n}{\binom {n}{r}}x^{r}}$

## Trigonometry

### Arc length

${\displaystyle l=r\theta \,\!}$

### Sector area

${\displaystyle A={\tfrac {1}{2}}r^{2}\theta }$

### Trigonometric identities

${\displaystyle \tan {\theta }\equiv {\frac {\sin {\theta }}{\cos {\theta }}}}$

${\displaystyle \sin ^{2}{\theta }+\cos ^{2}{\theta }\equiv 1\,\!}$

## Indices and logarithms

### Laws of indices

${\displaystyle x^{m}\times x^{n}=x^{m+n}\,\!}$

${\displaystyle x^{m}\div x^{n}=x^{m-n}\,\!}$

${\displaystyle \left(x^{m}\right)^{n}=x^{mn}\,\!}$

${\displaystyle x^{0}=1\,\!}$ (for x ≠ 0)

${\displaystyle x^{-m}={\frac {1}{x^{m}}}\,\!}$

${\displaystyle x^{\frac {1}{n}}={\sqrt[{n}]{x}}\,\!}$

${\displaystyle x^{\frac {m}{n}}={\sqrt[{n}]{x^{m}}}\,\!}$

### Logarithms

${\displaystyle 10^{2}=100\Leftrightarrow \log _{10}{100}=2}$

${\displaystyle 10^{3}=1000\Leftrightarrow \log _{10}{1000}=3}$

${\displaystyle 2^{5}=32\Leftrightarrow \log _{2}{32}=5}$

${\displaystyle \log _{a}{b}=c\Leftrightarrow a^{c}=b}$

### Laws of logarithms

The sum of the logs is the log of the product.

${\displaystyle \log {x}+\log {y}=\log {xy}\,\!}$

The difference of the logs is the log of the quotient.

${\displaystyle \log {x}-\log {y}=\log {\left({\frac {x}{y}}\right)}}$

The index comes out of the log of the power.

${\displaystyle k\log {x}=\log {\left(x^{k}\right)}}$

## Differentiation

### Differentiating the sum or difference of two functions

${\displaystyle y=f(x)\pm g(x)\quad \therefore \quad {\frac {dy}{dx}}=f'(x)\pm g'(x)}$

## Integration

### Integrating axn

${\displaystyle \int ax^{n}\,dx={\frac {ax^{n+1}}{n+1}}+c\qquad {\mbox{ for }}n\neq -1\,\!}$

### Area under a curve

The area under the curve ${\displaystyle y=f(x)}$ between the limits ${\displaystyle x=a}$ and ${\displaystyle x=b}$ is given by

${\displaystyle A=\int _{a}^{b}y\,dx}$