Christmas Problems

[jonboy]

1) If you buy every item in the song, The Twelve Days of Christmas, how many items will you buy?

Here's the song if you forgot it: Link
Would it be the summation of all the numbers 1 through 12 inclusive (78)?

Or would every time it repeats the "5 golden rings" would I have to add it to the number of items?

 

[morson]

Interesting question.

The number, S, of the nth mentioned item that the true love sent is:

S(n) = -n^2 + 13n (only valid for n being an element of {1,2,3...10,11,12}

So the sum is simply:

S(1) + S(2) + S(3) + S(4) ... + S(12), a finite series.

 

[soroban]

Hello, jonboy!

If you buy every item in the song, The Twelve Days of Christmas,
how many items will you buy?

\(\displaystyle 78\) is the number of items on the 12th day (only).
. . The answer is considerably larger . . .

Look at the shopping list:

\(\displaystyle \begin{array}{ccccc}\text{1st day} & \; & 1 & \,=\, & 1\\\text{2nd day} & \; & 1\,+\,2 & = & 3\\ \text{3rd day} & \; & 1\,+\,2\,+\,3 & = & 6\\\text{4th day} & \; & 1\,+\,2\,+\,3\,+\,4 & = & 10\\\vdots & \; & \vdots & & \vdots\end{array}\)

We are summing "triangular numbers".

The sum of the first \(\displaystyle n\) triangular numbers is: \(\displaystyle \:S(n)\:=\:\frac{n(n\,+\,1)(n\,+\,2)}{6}\)

Therefore: \(\displaystyle \:S(12)\;=\;\frac{12\cdot13\cdot14}{6}\;=\;364\)

 

[Denis]

[Therefore: \(\displaystyle \:S(12)\;=\;\frac{12\cdot13\cdot14}{6}\;=\;364\)

...and the Department Stores get richer...