Chapter 12.2, Problem 67E

The Twelve Days of Christmas In the well-known song “The Twelve Days of Christmas,” a person gives his sweet-heart k gifts on the kth day for each of the 12 days of Christmas. The person also repeats each gift identically on each subsequent day. Thus on the 12th day the sweetheart receives a gift for the first day, 2 gifts for the second, 3 gifts for the third, and so on. Show that the number of gifts received on the 12th day is a partial sum of an arithmetic sequence. Find this sum.

To determine

To Show: The number of gifts received on the 12^th day is a partial of an arithmetic sequence and find the sum.

Answer to Problem 67E

The number of gifts received on the 12^th day is a partial sum $S_k$ of an arithmetic sequence is $k=12a+66d$ and the sum is 78.

Explanation of Solution

Given:

The sequence of the gifts received by the sweat heart on 12th day is, $a_{12}=1+2+3+\mathrm{...}+12$.

Calculation:

Number of gifts is k on kth day for each of the 12 day of the Christmas to his sweetheart.

The total number of gifts $S_k$ is k.

The formula to find the sum of the term $S_n$ of the arithmetic progression shown below,

$S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$ (1)

Substitute 12 for n in equation (1),

$\begin{array}{c}{S}_{12}=\frac{12}{2}\left[2a+\left(12-1\right)d\right]\\ k=6\left[2a+11d\right]\\ k=12a+66d\end{array}$

Since, the person repeats the identical gifts on 12th day, 1 for the first day, 2 gifts for the second, 3 gifts for the third and so on.

The arithmetic progression formed through the gifts on 12th day provided by the person to his sweat heart is, $a_{12}=1+2+3+\mathrm{...}+12$.

Now, find the $a_{12}$ for the above arithmetic sequence,

Substitute 12 for n in equation (1),

$\begin{array}{c}{S}_{12}=\frac{12}{2}\left[2a+\left(12-1\right)d\right]\\ =6\left[2a+11d\right]\\ =12a+66d\end{array}$

Thus, the number of gifts received on the 12^th day is a partial sum $S_k$ of an arithmetic sequence is $k=12a+66d$.

The total number of gifts received on 12 th day is,

$a_{12}=12a+66d$ (2)

The given sequence is, $\text{Sequence}=1,2,3,\mathrm{...},12$

Substitute 1 for a, 1 for d in equation (1),

$\begin{array}{c}{S}_{12}=12\cdot 1+66\cdot 1\\ =12+66\\ =78\end{array}$

Thus, the sum of the gifts received on the 12 th day is 78.