Mathematical help with Wheatstone bridge
First you should know that the book answer is only an approximation. The exact equations are complex and nonlinear. 

If was going to solve this precisely, I would calculate Thevenin equivalent of both sides, which have a series resistance of 50||1000 on one side and (50 + a)|| 1000 on the other side, with 100 ohms between them. Then go and calculate the delta in Thevenin voltage to get the change in resistance for an exact 5uA through the galvo.

However we can quickly approximate if we assume constant current in the legs of the bridge of about 10mA and that gives you the change in Thevenin voltage. Simplify the total resistance and the book answer falls out. It's in error by maybe 5%, but close enough. 

The approach is critical in this kind  of question- it's almost a trick question because if you take a perfectionist approach you'll waste a lot of time and maybe have difficulty in an exam situation- but real life is like that too- if you take the wrong approach you need to recognize it as quickly as possible and abandon the overly complex approach.