A-level Mathematics/AQA/MPC2

Sequences and series

Notation

$u_n$ — the general term of a sequence; the n^th 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: $S_n=u_1+u_2+u_3+\dots +u_n$

$\sum$ — the sum of

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

$n\to \mathrm{\infty }$ — n tends towards infinity (n gets bigger and bigger)

$|x|$ — 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 n^th term gets closer to a finite number, L, as n approaches infinity. L is called the limit of the sequence:

$\text{As }n\to \mathrm{\infty }\text{, }{u}_n\to L$

Another way of denoting the same thing is:

$\underset{n\to \mathrm{\infty }}{\lim }{u}_n=L$

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 n^th 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.

$S_n=\frac{n}{2}[2a+(n-1)d]$

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

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:

$u_{n+1}=r{u}_n$

Expression for the nth term of an GP

$u_n=a{r}^{n-1}$

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

$S_{\mathrm{\infty }}=\sum _{n=1}^{\mathrm{\infty }}a{r}^{n-1}=\frac{a}{1-r}\text{where }-1<r<1$

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:

${}^n{C}_r=(\genfrac{}{}{0ex}{}{n}{r})=\frac{n!}{r!(n-r)!}$

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

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

Binomial expansion of (1+x)ⁿ

$(1+x{)}^n=1+(\genfrac{}{}{0ex}{}{n}{1})x+(\genfrac{}{}{0ex}{}{n}{2})x^2+(\genfrac{}{}{0ex}{}{n}{3})x^3+\dots +x^n$

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

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

Trigonometry

Arc length

$l=r\theta$

Sector area

$A=\frac{1}{2}{r}^2\theta$

Trigonometric identities

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

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

Indices and logarithms

Laws of indices

$x^m\times x^n=x^{m+n}$

$x^m÷x^n=x^{m-n}$

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

$x^0=1$ (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

$10^2=100\iff {\log }_{10}100=2$

$10^3=1000\iff {\log }_{10}1000=3$

$2^5=32\iff {\log }_{2}32=5$

${\log }_{a}b=c\iff a^c=b$

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.

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

Differentiation

Differentiating the sum or difference of two functions

$y=f(x)\pm g(x)\therefore \frac{dy}{dx}=f^{\prime }(x)\pm g^{\prime }(x)$

Integration

Integrating axⁿ

$\int a{x}^{n}dx=\frac{a{x}^{n+1}}{n+1}+c\text{ for }n\ne -1$

Area under a curve

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

$A=\underset{a}{\overset{b}{\int }}ydx$