Formulae
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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
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Let α {\displaystyle \alpha \,} and β {\displaystyle \beta \,} be the roots of a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} . Then, 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 \,} and γ {\displaystyle \gamma \,} be the roots of a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} . Then, ∑ 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 }
Matrices
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( A B ) − 1 = B − 1 A − 1 {\displaystyle \mathbf {(AB)^{-1}} =\mathbf {B^{-1}A^{-1}} }