My approach would be to start with the given relationships:
for the A.P. property
for the G.P. property
for the H.P property.
Tidy up as far as possible, then substitute for and to get an equation demonstrating the result, which should be reducible to:
I've just tried it on paper but haven't got the result I was expecting - I probably made a mistake in my algebra (easy to do). Try it out, see what you get, post your working on here and see what you get. (I won't be on line again today as from 20 minutes from now till about 12 hours time, unfortunately.)
Edit: Yes, I've tried it again, and the above approach does work.