1998 HSC AM-GM inequality induction proof

[porcupinetree]

I was doing the 1998 HSC for Extension 2 and this question popped up. Parts (i) and (ii) were easy but (iii) is a lot more tricky. The marking notes explicitly state that the inequality in (ii) must be used, but I can't seem to figure out exactly how it should be used. Can someone help with this induction proof? Thanks.

Interesting trivia: Only 4 people successfully completed this proof in the actual 1998 HSC (according to the marking notes).

 

[Carrotsticks]

It's a nice application of Young's Inequality to prove out AM/GM.

 

[Drsoccerball]

How do you get rid of the k+1 at the bottom ive tried everything...

 

[Carrotsticks]

How do you get rid of the k+1 at the bottom ive tried everything...

I used the inequality from the previous part.

 

[Drsoccerball]

I used the inequality from the previous part.

I did that and i had the answer with a k+1 at the bottom :/...