Too Many Nines

${ \underbrace { 99 }_{ 2 } }^{ 2 }=9801\quad \rightarrow \quad The\quad sum\quad is\quad 9+(8+1)\quad =\quad 2\times 9\\ { \underbrace { 999 }_{ 3 } }^{ 2 }=998001\quad \rightarrow \quad The\quad sum\quad =\quad 3\times 9\\ { \underbrace { 99...9 }_{ 1001 } }^{ 2 }\quad \rightarrow \quad The\quad sum\quad =\quad 9\times 1001$

it follows a cycle since the square of any number of nines will end up ( ( n - 1 )9x + 9 ), then 1000x9 + 9 = 9009