Longer and Longer

Tape measure on a plain background
You may wish to take a look at Measuring with Rods before trying this problem.
Charlie has been exploring the lengths he can measure with two different rods:
Alison has been exploring the lengths she can measure if she adds a third rod:

Is there only one set of three rods that will allow Alison to measure the largest number of consecutive lengths?

What if she could use a set of four rods?... Or five? ...

Is there a way to determine the ideal lengths for any number of rods?

Is there a quick way to work out how far a set of rods will allow you to measure?


What would happen if you could only add lengths together, and couldn't subtract?