Question 9 - A-Level Maths

Show that $\alpha ^ { 3 } + \beta ^ { 3 } = ( \alpha + \beta ) ^ { 3 } - 3 \alpha \beta ( \alpha + \beta )$.
The quadratic equation $x ^ { 2 } - 5 x + 7 = 0$ has roots $\alpha$ and $\beta$. Find a quadratic equation with roots $\alpha ^ { 3 }$ and $\beta ^ { 3 }$.
Show that $\frac { 2 } { r } - \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$.
Hence find an expression, in terms of $n$, for $$\sum _ { r = 1 } ^ { n } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$$
Hence write down the value of $\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$.
Given that $\sum _ { r = N + 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { 7 } { 10 }$, find the value of $N$.