Formulae

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)

Series

• ${\displaystyle \sum _{r=1}^{n}r={\frac {1}{2}}n(n+1)}$
• ${\displaystyle \sum _{r=1}^{n}r^{2}={\frac {1}{6}}n\left(2n+1\right)\left(n+1\right)}$ †
• ${\displaystyle \sum _{r=1}^{n}r^{3}={\frac {1}{4}}n^{2}\left(n+1\right)^{2}}$ †

Roots of Polynomials

• Let ${\displaystyle \alpha \,}$ and ${\displaystyle \beta \,}$ be the roots of ${\displaystyle ax^{2}+bx+c=0}$. Then, ${\displaystyle \alpha +\beta =-{\frac {b}{a}},\quad \alpha \beta ={\frac {c}{a}}}$
• Let ${\displaystyle \alpha ,\beta \,}$ and ${\displaystyle \gamma \,}$ be the roots of ${\displaystyle ax^{3}+bx^{2}+cx+d=0}$. Then, ${\displaystyle \sum \alpha =-{\frac {b}{a}},\quad \sum \alpha \beta ={\frac {c}{a}},\quad \alpha \beta \gamma =-{\frac {d}{a}}}$

Where: ${\displaystyle \sum \alpha =\alpha +\beta +\gamma }$

And: ${\displaystyle \sum \alpha \beta =\alpha \beta +\alpha \gamma +\beta \gamma }$

Matrices

• ${\displaystyle \mathbf {(AB)^{-1}} =\mathbf {B^{-1}A^{-1}} }$