proofs

hi guys this might be weird but could somebody please explain how to use AM-GM in questions, maybe with some examples? idk why but this particular part of the content we've learnt so far just does not absorb into my brain, I can prove the inequality itself, but the moment I have to use it (I can...

I got part a ) For b) I managed to prove k^k < 1/4(k+1)^k+1 but i am struggling to prove the middle bit for some reason which shouldnt be that hard but idk why im struggling lol