For a natural number $m$, there are blocks in the shape of identical cubes stacked with 1 block in column 1, 2 blocks in column 2, 3 blocks in column 3, $\cdots$, and $m$ blocks in column $m$. The following procedure is repeated until there are no columns with an even number of blocks remaining. For each column with an even number of blocks, remove $\frac { 1 } { 2 }$ of the blocks in that column from the column. After completing all block removal procedures, let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$. For example, $f ( 2 ) = 2 , f ( 3 ) = 5 , f ( 4 ) = 6$. $$\lim _ { n \rightarrow \infty } \frac { f \left( 2 ^ { n + 1 } \right) - f \left( 2 ^ { n } \right) } { f \left( 2 ^ { n + 2 } \right) } = \frac { q } { p }$$ Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]
For a natural number $m$, there are blocks in the shape of identical cubes stacked with 1 block in column 1, 2 blocks in column 2, 3 blocks in column 3, $\cdots$, and $m$ blocks in column $m$. The following procedure is repeated until there are no columns with an even number of blocks remaining.
For each column with an even number of blocks, remove $\frac { 1 } { 2 }$ of the blocks in that column from the column.
After completing all block removal procedures, let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$.\\
For example, $f ( 2 ) = 2 , f ( 3 ) = 5 , f ( 4 ) = 6$.
$$\lim _ { n \rightarrow \infty } \frac { f \left( 2 ^ { n + 1 } \right) - f \left( 2 ^ { n } \right) } { f \left( 2 ^ { n + 2 } \right) } = \frac { q } { p }$$
Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]