James Ruse 2006 Trial Question (also asked in Baulkham Hills 2023 Trial) - does part (ii) and (iii) actually work? (1 Viewer)

[lyounamu]

Hi everyone. Someone asked me this question from BHS 2023 Trial so I thought I would give this Q a go and I remembered that I did this very question long long time ago from JRAH Trial as well. Part (i) was a stock standard question but the results from part (ii) and part (iii) are the ones that I wanted to get everyone's views on.


My question is, do the results from part (ii) and (iii) legitimately hold? For part (iii), I can find plenty of counter examples like 1+1+1+0.9<3.4+0.2+0.2+0.2 that invalidates the given outcome but wanted to double and triple check with everyone. JRAH conveniently omits solutions for the entire question and BHS has solutions for part (ii) and (iii) but I don't agree with the inequality treatment in part (ii). I wont even go to part (iii) solutions because it relies on the use of part (ii) solutions. Attached below is the solution from BHS.

 

[member 6003]

You are right, the reason why is because whoever made this paper forgot to copy one of the conditions, . This should fix the counter example you have I believe. This makes the reasoning they give in ii) valid since actually holds true regardless of order. The way they get the given equation is using Abel's Theorem for summation by parts which looks like this:

you can find a proof for it online somewhere.

 

[lyounamu]

Thanks for your response. Yep, the major issue that I had with part ii solution was that the inequality wouldn't hold if b_(n-1)>b_n but it could be resolved by introducing that condition that you mentioned. Thanks for that.

 

[lyounamu]

Actually, BHS did say b_n>b_(n-1) and so on. So it's all good.