back tire keeps going down I have just bought a new bike and had it about a month now. I don't really know much about bikes but I do know how to repair an inner tube.

I have had one puncture and just bought a new inner tube. It worked for a few days and then went down again so this time I got a puncture repair kit. I took the inner tube out to check for any punctures and found nothing. I put it in water and pumped it up and there were no bubbles. I went around the entire tube pushing it next to my ear and heard nothing. I checked the tire to see if anything was wrong - there was nothing that I could see. 

So I put it all back together and took it for a ride and got about half a mile and the tire was flat. I pumped it up and got about another half a mile and it was flat again.

Do you have any suggestions for me?
It could be an bad valve. I had a run of flats on one bike and was sure it was something wrong in the wheel. Turns out I hadn't replaced the tubes in a long time so the tubes I installed were also old. Both failed where the stem meets the tube.  The pressure needed to be high and it took about 45mins to go flat.