Question 3 - A-Level Maths

Show that $$( r + 1 ) ! - ( r - 1 ) ! = \left( r ^ { 2 } + r - 1 \right) ( r - 1 ) !$$
Hence show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + r - 1 \right) ( r - 1 ) ! = ( n + 2 ) n ! - 2$$ (4 marks) The cubic equation $$z ^ { 3 } - 2 z ^ { 2 } + k = 0 \quad ( k \neq 0 )$$ has roots $\alpha , \beta$ and $\gamma$.
1. Write down the values of $\alpha + \beta + \gamma$ and $\alpha \beta + \beta \gamma + \gamma \alpha$.
2. Show that $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 4$.
3. Explain why $\alpha ^ { 3 } - 2 \alpha ^ { 2 } + k = 0$.
4. Show that $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 8 - 3 k$.
Given that $\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } = 0$ :
1. show that $k = 2$;
2. find the value of $\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 }$.