1. ## Word problem

In the well-know song "the twelve days of Christmas" a person gives his or her true love k gifts on the k-th day for each of the 12 days of Christmas and repeats each gift identically on the subsequent day. Thus the twelfth day the true love receives one gift plus two gifts plus three gifts for the third day, and so on.

2. Hello,

Originally Posted by gumi
In the well-know song "the twelve days of Christmas" a person gives his or her true love k gifts on the k-th day for each of the 12 days of Christmas and repeats each gift identically on the subsequent day. Thus the twelfth day the true love receives one gift plus two gifts plus three gifts for the third day, and so on.
On the 12th day, the true love receives 1+2+3+...+12, do you agree ?

Remember the formula from which Gauss gifted a sample to his teacher :

$\displaystyle 1+2+\dots+n=\frac{n(n+1)}{2}$

This is more tricky..

Let $\displaystyle a_n$ be the number of presents received on the n-th day.

From the formula above, $\displaystyle a_n=\frac{n(n+1)}{2}$.

We want to calculate $\displaystyle S=a_1+a_2+\dots+a_{12}=\sum_{n=1}^{12} \frac{n(n+1)}{2}$

\displaystyle \begin{aligned} S &=\frac 12 \sum_{n=1}^{12} n^2+n \\ &=\frac 12 \sum_{n=1}^{12} n^2+\frac 12 \sum_{n=1}^{12} n \end{aligned}

You already know what the second term is equal to..

Now, there is another formula you should know :

$\displaystyle \sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}$

And normally, it's all done

3. Hello, gumi!

Are we expected to derive the necessary formulas?
. . Or is simple addition sufficient?

In the well-known song "the twelve days of Christmas", a person gives his or her true love
$\displaystyle k$ gifts on the $\displaystyle k^{th}$ day for each of the 12 days of Christmas and repeats each gift identically
on each subsequent day. Thus the twelfth day the true love receives one gift plus two gifts
On the 12th day: .$\displaystyle 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 \;=\;78$ gifts.
Total: .$\displaystyle 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 \;=\;364$ gifts.