Shared Pie Ⅱ

Calculus Level 4

At a party with $n$ people, there is a pie. The first guest gets $\frac 1n$ of the pie, the second gets $\frac 2n$ of what is left, the third gets $\frac 3n$ of what is left and so on.

If the $k_n$ -th guest gets the largest piece of all $n$ guests and $\displaystyle\lim_{n\to\infty}\frac {k_n}{n^\alpha}=\frac 1\beta,$ what's the value of $\alpha+\beta?$

Try this problem first.

The answer is 1.5.

1 solution

Parth Sankhe
Nov 24, 2018

As discussed in the earlier problem, the $kth$ guest will get the most pie, where k is the positive root of $k^2-k-n=0$

(I have denoted $\alpha$ as $a$ and $\beta$ as $b$ )

lim $\frac {-1+\sqrt{1+4n}}{2n^a}$

Since $n\rightarrow ∞$ , $4n+1≈4n$ , and $2√n-1≈2√n$

lim $\frac {2√n}{2n^a}$

For the limit to exist, $a=0.5$

Therefore, lim $\frac {2}{2}=1$

Hence $b=1$ .

$a+b=0.5+1=1.5$

I solved this in pretty much the same way and wondered why the answer asks for $\frac{1}{b}$ when $b=1$ anyway.

The reason is, the limit also exists for $a>0.5$ but as this limit is then $0$ , it would make $b$ undefined.

Jeremy Galvagni - 2 years, 6 months ago

Exactly! 😊

Brian Lie - 2 years, 6 months ago

I think you mean $a>0.5$ , and yeah I thought about that too, thanks for telling me the reason!

Parth Sankhe - 2 years, 6 months ago

Shared Pie Ⅱ

The answer is 1.5.

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