By the end of this module you will be expected to have learnt the following formulae:
Formulae marked † are in the standard OCR Maths Data book (as of 2010)
∑
r
=
1
n
r
=
1
2
n
(
n
+
1
)
{\displaystyle \sum _{r=1}^{n}r={\frac {1}{2}}n(n+1)}
∑
r
=
1
n
r
2
=
1
6
n
(
2
n
+
1
)
(
n
+
1
)
{\displaystyle \sum _{r=1}^{n}r^{2}={\frac {1}{6}}n\left(2n+1\right)\left(n+1\right)}
∑
r
=
1
n
r
3
=
1
4
n
2
(
n
+
1
)
2
{\displaystyle \sum _{r=1}^{n}r^{3}={\frac {1}{4}}n^{2}\left(n+1\right)^{2}}
Roots of Polynomials
edit
Let
α
{\displaystyle \alpha \,}
β
{\displaystyle \beta \,}
a
x
2
+
b
x
+
c
=
0
{\displaystyle ax^{2}+bx+c=0}
α
+
β
=
−
b
a
,
α
β
=
c
a
{\displaystyle \alpha +\beta =-{\frac {b}{a}},\quad \alpha \beta ={\frac {c}{a}}}
Let
α
,
β
{\displaystyle \alpha ,\beta \,}
γ
{\displaystyle \gamma \,}
a
x
3
+
b
x
2
+
c
x
+
d
=
0
{\displaystyle ax^{3}+bx^{2}+cx+d=0}
∑
α
=
−
b
a
,
∑
α
β
=
c
a
,
α
β
γ
=
−
d
a
{\displaystyle \sum \alpha =-{\frac {b}{a}},\quad \sum \alpha \beta ={\frac {c}{a}},\quad \alpha \beta \gamma =-{\frac {d}{a}}}
Where:
∑
α
=
α
+
β
+
γ
{\displaystyle \sum \alpha =\alpha +\beta +\gamma }
And:
∑
α
β
=
α
β
+
α
γ
+
β
γ
{\displaystyle \sum \alpha \beta =\alpha \beta +\alpha \gamma +\beta \gamma }
(
A
B
)
−
1
=
B
−
1
A
−
1
{\displaystyle \mathbf {(AB)^{-1}} =\mathbf {B^{-1}A^{-1}} }
