On the coordinate plane, for a natural number $n$, the coordinates of point $\mathrm { P } _ { n }$ are $\left( n , 3 ^ { n } \right)$ and the coordinates of point $\mathrm { Q } _ { n }$ are $( n , 0 )$. Let $a _ { n }$ be the area of the quadrilateral $\mathrm { P } _ { n } \mathrm { Q } _ { n + 1 } \mathrm { Q } _ { n + 2 } \mathrm { P } _ { n + 1 }$. When $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { a _ { n } } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (where $p$ and $q$ are coprime natural numbers) [4 points]
On the coordinate plane, for a natural number $n$, the coordinates of point $\mathrm { P } _ { n }$ are $\left( n , 3 ^ { n } \right)$ and the coordinates of point $\mathrm { Q } _ { n }$ are $( n , 0 )$.\\
Let $a _ { n }$ be the area of the quadrilateral $\mathrm { P } _ { n } \mathrm { Q } _ { n + 1 } \mathrm { Q } _ { n + 2 } \mathrm { P } _ { n + 1 }$. When $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { a _ { n } } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (where $p$ and $q$ are coprime natural numbers) [4 points]