[dba-SQLServer] Execution statistics

If you an add an index that improves the execution time, but also increases
the logical reads -- is that a problem for the future? It seems to me that
as you add data those page hits are going to eventually slow you down,
despite the index. Am I analyzing this correctly? 
 
Or, do indexes always increase the logical reads -- that would make sense to
me too. 
 
Susan H.