The easiest thing is to copy and paste; the change from `c3:al3` to `c4:al4`, etc, will happen automatically. I would proceed as follows:

1. Enter the formula `=average(c3:al3)` in `AN3`. 
2. Select cell `AN3`, press <kbd>CTRL</kbd>+<kbd>C</kbd> (or <kbd>CMD</kbd>+<kbd>C</kbd>
 on Mac) to copy it (the cell itself, not the formula from the formula line).
3. Select the range `AN3:AN562` (or whichever range you want to paste to). 
4. Press <kbd>CTRL</kbd>+<kbd>V</kbd> (or <kbd>CMD</kbd>+<kbd>V</kbd> on Mac). 

The formula that will actually appear in the cells will have appropriate row numbers.

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A mouse-based way to do the same is to drag the formula down using the dot in the bottom right corner of the cell.