A-level Mathematics/AQA/MPC2

Notation
$$ u_n \,\!$$ — the general term of a sequence; the nth term

$$ a \,\!$$ — the first term of a sequence

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

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

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

$$ S_n \,\!$$ — the sum to n terms: $$ S_n = u_1 + u_2 + u_3 + \ldots + u_n \,\!$$

$$ \sum \,\!$$ — the sum of

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

$$ n \rightarrow \infty \,\!$$ — n tends towards infinity (n gets bigger and bigger)

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

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:

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

Another way of denoting the same thing is:

$$ \lim_{n \to \infty}u_n = L \,\!$$

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

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

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:

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

Expression for the nth term of an AP
$$ u_n = a + (n-1)d \,\!$$

Formulae for the sum of the first n terms of an AP
The sum of an arithmetic progression is called an arithmetic series.

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

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

Formulae for the sum of the first n natural numbers
The natural numbers are the positive integers, i.e. 1, 2, 3…

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

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:

$$ u_{n+1} = ru_n \,\!$$

Expression for the nth term of an GP
$$ u_n = ar^{n-1} \,\!$$

Formula for the sum of the first n terms of a GP
$$ S_n = a \left ( \frac{1-r^n}{1-r} \right ) \,\!$$

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

Formula for the sum to infinity of a GP
$$ S_\infty = \sum_{n=1}^\infty ar^{n-1} = \frac{a}{1-r} \qquad \mbox{where } -1 < r < 1 \,\!$$

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 $$(a+b)^n$$.

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

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

where $$n! = 1 \times 2 \times 3 \times \ldots \times n$$

$$n!$$ is called n factorial. By definition, $$0!=1$$.

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

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

$$(1+x)^n=\sum_{r=0}^n\binom{n}{r}x^r$$

Arc length
$$l = r \theta \,\!$$

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

Trigonometric identities
$$\tan{\theta} \equiv \frac{\sin{\theta}}{\cos{\theta}}$$

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

Laws of indices
$$ x^m \times x^n = x^{m+n} \,\!$$

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

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

$$ x^0 = 1 \,\!$$ (for x ≠ 0)

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

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

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

Logarithms
$$10^2 = 100 \Leftrightarrow \log_{10}{100} = 2 $$

$$10^3 = 1000 \Leftrightarrow \log_{10}{1000} = 3 $$

$$2^5 = 32 \Leftrightarrow \log_{2}{32} = 5 $$

$$\log_a{b} = c \Leftrightarrow a^c = b $$

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

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

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

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

The index comes out of the log of the power.

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

Differentiating the sum or difference of two functions
$$ y = f(x) \pm g(x) \quad \therefore \quad \frac{dy}{dx} = f'(x) \pm g'(x)$$

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

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

$$ A = \int_a^b y \, dx $$