Beyond the 12 Days of Christmas

Algebra Level 3

In “The Twelve Days of Christmas” song, the singer’s true love gives her $1$ gift the first day, $2 + 1$ gifts the second day, $3 + 2 + 1$ gifts the third day, and so on.

Not to be outdone, the singer has a new suitor who decides to give her $1^2$ gifts the first day, $2^2$ gifts the second day, $3^2$ gifts the third day, and so on.

Let $T(n)$ be the total number of accumulated gifts given by the singer’s true love after $n$ days, and $S(n)$ be the total number of accumulated gifts given by the singer’s new suitor after $n$ days.

Find $\displaystyle \lim_{n \to \infty} \frac{S(n)}{T(n)}$ .

The answer is 2.

1 solution

Gabriel Chacón
Dec 23, 2018

We can answer the question without finding the expressions for $T(n)$ and $S(n)$ .

Let's consider what the singer receives the $n$ -th day:

From her true love: $t_n=1+2+\ldots+(n-1)+n=\dfrac{n(n+1)}{2}^*$
From the new suitor: $s_n=n^2$

$\displaystyle \lim_{n \to \infty}\dfrac{s_n}{t_n}=\lim_{n \to \infty}\dfrac{n^2}{\frac{n^2+n}{2}}=\lim_{n \to \infty}2\cdot \dfrac{1}{1+\frac{1}{n}}=2$ , and this is also true for the quotient of the sums, as their terms grow indefinitely and make the initial deviation from this ratio fade away. $\displaystyle \lim_{n \to \infty}\dfrac{S(n)}{T(n)}=\boxed 2$

* This follows from observing that $1+n=2+(n-1)=3+(n-2)=\ldots$ and that there are $n/2$ of these couples.

Great solution!

David Vreken - 2 years, 5 months ago

Great problem... And solution!

My only comment is that $n$ isn't mentioned until the very last line.

Although perhaps intuitive, perhaps it should be defined and/or mentioned earlier in the problem?

Geoff Pilling - 2 years, 5 months ago

You're right, when I wrote the problem I accidentally used x and n interchangeably. I edited the problem and changed all the x's to n's.

David Vreken - 2 years, 5 months ago

Ah yes... much cleaner! :)

Geoff Pilling - 2 years, 5 months ago

Beyond the 12 Days of Christmas

The answer is 2.

1 solution

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