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1. Number and Algebra
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1. Number and Algebra
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1. Number and Algebra
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Sub-topics
•
Arithmetic sequences and series (1.2)
•
Use of the formulae for the
n
t
h
n^{th}
n
t
h
term and the sum of the first
n
n
n
terms of the sequence.
•
Use of sigma notations of sums of arithmetic sequences.
•
Applications
•
Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.
•
Geometric sequences and series. (1.3)
•
Use of the formulae for the
n
t
h
n^{th}
n
t
h
term and the sum of the first
n
n
n
terms of the sequence.
•
Use of sigma notation for the sums of geometric sequences.
•
Applications
•
Financial applications of geometric sequences and series: ie) compound interest, annual depreciation. (1.4)
Δ
θ
=
Arc
Radius
Δ
θ
=
Δ
s
r
and
Δ
θ
⋅
r
=
Δ
s
\begin{aligned} &\Delta \theta=\frac{\text { Arc }}{\text { Radius }}\\ &\Delta \boldsymbol{\theta}=\frac{\Delta \boldsymbol{s}}{r} \text { and } \boldsymbol{\Delta} \boldsymbol{\theta} \cdot \boldsymbol{r}=\boldsymbol{\Delta} \boldsymbol{s} \end{aligned}
Δ
θ
=
Radius
Arc
Δ
θ
=
r
Δ
s
and
Δ
θ
⋅
r
=
Δ
s
Sequence
a
n
a_n
a
n
is an enumerated collection of objects in which repetitions are allowed and order matters.
There are two sequences to be aware of:
arithmetic
and
geometric
.
1.2 Sequences and Series
•
Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. (1.6)
•
The symbols and notation for equality and identity.
•
Proof by mathematical induction (AHL 1.15)
•
Proof by contradiction
•
Use of a counterexample to show that a statement is not always true.
1.3 Proof and Reasoning
•
The binomial theorem: expansion of
(
a
+
b
)
n
(a + b)^n
(
a
+
b
)
n
,
n
∈
N
n ∈ N
n
∈
N
.
(1.9)
•
Use of Pascal’s triangle and
n
C
r
^nC_r
n
C
r
.
•
Counting principles, including permutations and combinations. (AHL 1.10)
•
Extension of the binomial theorem to fractional and negative indices, ie
(
a
+
b
)
n
(a + b)^n
(
a
+
b
)
n
,
n
∈
Q
n ∈ Q
n
∈
Q
.
Use of Pascal’s triangle
Figure 1.4.1
Pascal’s triangle
For the binomial expansion of
(
a
x
+
b
)
n
(ax+b)^n
(
a
x
+
b
)
n
where
n
∈
N
n∈ N
n
∈
N
:
•
The powers of ax decrease by 1, and on the other hand, the powers of b increases by 1
•
The expansion contains
n
+
1
n+1
n
+
1
terms
•
The sum of the exponents of
a
x
ax
a
x
and
b
b
b
in each term equals
n
n
n
•
The coefficients of the terms correspond to the
n
n
n
th row of Pascal’s triangle
Binomial series
Binomial expansion
of two terms
(
a
x
+
b
)
n
(ax+b)^n
(
a
x
+
b
)
n
for
n
∈
Z
+
n∈Z^+
n
∈
Z
+
is given by:
1.4 Binomial Theorems and Combinatorics
•
The number
i
i
i
, where
i
2
=
−
1
i^2=-1
i
2
=
−
1
.
(1.12)
•
Cartesian form
z
=
a
+
b
i
z = a + bi
z
=
a
+
bi
; the terms real part, imaginary part, conjugate, modulus and argument.
•
The complex plane.
•
Modulus–argument (polar) form:
z
=
r
(
cos
θ
+
i
sin
θ
)
=
r
c
i
s
θ
z = r(\cosθ + i\sinθ) = rcisθ
z
=
r
(
cos
θ
+
i
sin
θ
)
=
rc
i
s
θ
(1.13)
•
Euler form:
z
=
r
e
i
θ
z = re^{iθ}
z
=
r
e
i
θ
•
Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.
•
Complex conjugate roots of quadratic and polynomial equations with real coefficients.
(1.14)
•
De Moivre’s theorem and its extension to rational exponents.
•
Powers and roots of complex numbers.
1.5 Complex Numbers (HL)
1.6 Linear Equations
done
1.6 Linear Equations (HL)
1.6 Linear Equations (HL)
