A-Level Maths OCR B (MEI) H640

Understand and use the binomial expansion of $(a + bx)^n$ where $n$ is a positive integer.

Know the notations $n!$ and $_nC_r$ and that $_nC_r$ is the number of ways of selecting $r$ distinct objects from $n$.

The meaning of the term factorial.

$n$ a positive integer.

Link to binomial probabilities.

Notation:

• $_nC_r = \dfrac{n!}{r!(n-r)!}$
• $n! = 1×2×3×...×n$
• $_nC_0 = _nC_n = 1$
• $0! = 1$
• $^nC_r, \dbinom{n}{r}$

Excludes: $_nC_r$ will only be used in the context of binomial expansions and binomial probabilities.

Use the binomial expansion of $(1 + x)^n$ where $n$ is any rational number.

For $|x| < 1$ when $n$ is not a positive integer.

Excludes: General term.

Be able to write $(a + bx)^n$ in the form $a^n\Bigg(1+\dfrac{bx}{a}\Bigg)^n$ and hence expand $(a + bx)^n$.

*$\Big|\dfrac{bx}{a}\Big| < 1$ when $n$ is not a positive integer.

Excludes: Proof of convergence.*

Be able to use binomial expansions with $n$ rational to find polynomials which approximate $(a + bx)^n$.

Includes finding approximations to rational powers of numbers.

Know what a sequence of numbers is and the meaning of finite and infinite with reference to sequences.

Be able to generate a sequence using a formula for the $k$th term, or a recurrence relation of the form $a_{k+1} = f(a_k)$.

e.g. $a_k = 2 + 3k$;

$a_{k+1} = a_k + 3$ with $a_1 = 5$.

Notation:

• $k$th term: $a_k$

Know that a series is the sum of consecutive terms of a sequence.

Starting from the first term.

Understand and use sigma notation.

Notation:

• $\displaystyle\sum_{r=1}^n{r} = 1+2+...+n$

Be able to recognise increasing, decreasing and periodic sequences.

Know the difference between convergent and divergent sequences.

Including when using a sequence as a model or when using numerical methods.

Notation:

• Limit to denote the value to which a sequence converges.

Excludes: Formal tests for convergence.

Understand and use arithmetic sequences and series.

The term arithmetic progression (AP) may also be used for an arithmetic sequence.

Notation:

• First term, $a$
• Last term, $l$
• Common difference, $d$

Be able to use the standard formulae associated with arithmetic sequences and series.

The $n$th term, the sum to $n$ terms.

Including the sum of the first $n$ natural numbers.

Notation:

• $S_n$

Understand and use geometric sequences and series.

The term geometric progression (GP) may also be used for a geometric sequence.

Notation:

• First term, $a$
• Common ratio, $r$

Be able to use the standard formulae associated with geometric sequences and series.

The $n$th term, the sum to $n$ terms.

Notation: $S_n$

Know the condition for a geometric series to be convergent and be able to find its sum to infinity.

$S_∞ = \dfrac{a}{1-r}, |r| < 1$

Be able to use sequences and series in modelling.