Quickest way to Read Specific Line from Multiple Files one Line at a time

I have a text-based database that represents logs, sorted by timestamp. For testing purposes my database has approximately 10,000 lines but this number can be larger. It is of the format:

primary_key, source_file, line_num
1, cpu.txt, 2
2, ram.txt, 3
3, cpu.txt, 3

I query the database and as I read the results I want to add the actual data to a string which I can then display. Actual data in the above example would be the contents of line 2 from cpu.txt, followed by the contents of line 3 from ram.txt, etc. The line contents can be quite long.

An important note is that the line numbers per file are all in order. That is to say, the next time I encounter a cpu.txt entry in the database it will have line 4 as the line number. However, I might see a cpu.txt entry only after thousands of other entries from ram.txt, harddrive.txt, graphics.txt, etc.

I have thought about using something along the lines of the following code:

StringBuilder odbcResults = new StringBuilder();
OdbcDataReader dbReader = com.ExecuteReader();  // query database
while (dbReader.Read())
{
   string fileName = dbReader[1].ToString(); // source file
   int fileLineNum = int.Parse(dbReader[2].ToString());  // line number in source file

   odbcResults.Append(File.ReadLines(fileName).Skip(fileLineNum).First());
}

However, won't File.ReadLines() want to dispose of its TextReader after every iteration? Not very efficient?

I also had this idea, keeping a StreamReader for every file that I need to read in a Dict:

Dictionary<string, StreamReader> fileReaders = new Dictionary<string, StreamReader>();
StringBuilder odbcResults = new StringBuilder();
OdbcDataReader dbReader = com.ExecuteReader();
while (dbReader.Read())
{
   string fileName = dbReader[1].ToString(); // source file
   int fileLineNum = int.Parse(dbReader[2].ToString());  // line number in source file

   if (!fileReaders.ContainsKey(fileName))
   {
      fileReaders.Add(fileName, new StreamReader(fileName));
   }

   StreamReader fileReader = fileReaders[fileName];
   // don't have to worry about positioning? Lines consumed consecutively
   odbcResults.Append(fileReader.ReadLine());
}
// can't forget to properly Close() and Dispose() of all fileReaders

Do you agree with any of the above examples or is there an even better way?
For the second example I am running on the assumption that the StreamReader will remember its last position - I believe this is saved in the BaseStream.

I have read over How do I read a specified line in a text file?, Read text file at specific line, StreamReader and seeking (the first answer provides a link to a custom StreamReader class with positioning capabilities, but I only know the line number I need to be on, not an offset) but don't think they answer my question specifically.