#6.1
Understand and use the binomial expansion of \((a + bx)^n\) where \(n\) is a positive integer.
#6.2
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.]
#6.3
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.]
#6.4
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.]
#6.5
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.
#6.6
Know what a sequence of numbers is and the meaning of finite and infinite with reference to sequences.
#6.7
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: \(a_k\)
#6.8
Know that a series is the sum of consecutive terms of a sequence.
Starting from the first term.
#6.9
Understand and use sigma notation.
Notation:
\(\displaystyle\sum_{r=1}^n{r} = 1+2+...+n\)
#6.11
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.]
#6.12
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, \(d\)
#6.13
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: \(S_n\)
#6.14
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, \(r\)
#6.15
Be able to use the standard formulae associated with geometric sequences and series.
The \(n\)th term, the sum to \(n\) terms.
Notation: \(S_n\)
#6.16
Know the condition for a geometric series to be convergent and be able to find its sum to infinity.
\(S_∞ = \dfrac{a}{1-r}, |r| < 1\)
#6.17
Be able to use sequences and series in modelling.
Arithmetic sequences and series Geometric sequences and series