No . Well actually it depends on what type of nurse you want to be , you say adult nurse but are you looking to be a nurse in a hospital or at a doctors? Cus some of them reqiure much bigger tasks and responsibiliteis , if you;re looking to work at a doctors you will deal with injecting people , (will have to go on a course for that) and aching muscles etc all that primary kind of stuff but very vital and important and if you;re looking to be a nurse at a hospital you will be looking after patients with severe conditions etc , so you have to specific.
Yes you'll need a degree in Nursing Studies if you want to specialise in Geriatrics. A Diploma in Nursing is available, but this is an "entry level" nursing qualification and limits what you can actually specialise in.