Problem 63 - Algebra and Trigonometry

Question

The Twelve Days of Christmas In the well-known song "The Twelve Days of Christmas," a person gives his sweetheart \(k\) gifts on the \(k\)th 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 received on the 12th day is a partial sum of an arithmetic sequence. Find this sum.

Step-by-Step Solution

Answer

78 gifts are received on the 12th day.

Step 1: Understand the Pattern of Gift Giving

In this problem, the person gives gifts in an increasing sequence each day. On the 1st day, 1 gift is given; on the 2nd day, 1 + 2 gifts are given, and so on. Thus, on the 12th day, gifts are given from day 1 to day 12, such that gifts on the 12th day correspond to a partial sum of the gifts distributed over the 12 days.

Step 2: Determine the Total Number of Gifts for Each Day

To find the total gifts given on any particular day up to the 12th day, you need to add the gifts given on that day plus all previous days. Therefore, the given gifts on day k is the sum of gifts from day 1 to day k.

Step 3: Calculate the Cumulative Gifts on Each Day

For the 12th day, the total number of gifts is the sum of the arithmetic series 1 + 2 + 3 + ... + 12. The formula for the sum of the first n natural numbers is \[ S_n = \frac{n(n+1)}{2} \]where \( n = 12 \).

Step 4: Calculate the Sum of Gifts on the 12th Day Using the Formula

Substitute \( n = 12 \) into the formula:\[ S_{12} = \frac{12(12+1)}{2} = \frac{12 \times 13}{2} = 78 \]Thus, the number of gifts received on the 12th day is 78.

Key Concepts

partial sum

In arithmetic sequences, a partial sum involves adding together a certain number of its initial terms. The concept of partial sum helps us determine how many items are accumulated over a series of actions or time.

For instance, in the "Twelve Days of Christmas," by the 12th day, you want to calculate the sum of gifts given from day 1 to day 12. This is what is known as a partial sum. It's essentially the sum you get before you reach the very end of a sequence.

Here are the steps to understand partial sums in sequences:

• Identify the specific terms that need to be added. In this case, these are the number of gifts given each day.
• Add each term sequentially until reaching the specified day or step. For day 12, this is the sum of 1, 2, 3, ..., up to 12.
• The result is the partial sum, which provides the cumulative total at that point in the sequence.

When computing a partial sum, understanding its relation to the full sequence is important. It is key in many mathematical applications, such as calculating interest over time or managing resources in a project.

arithmetic series

An arithmetic series is quite simply the sum of the terms in an arithmetic sequence. An arithmetic sequence is a list of numbers where each term increases by a constant value, known as the common difference.

To form an arithmetic series, you add the terms of an arithmetic sequence together. For example, the sequence 1, 2, 3, ... up to 12 in the "Twelve Days of Christmas" can be summed to form an arithmetic series.

The approach to calculate an arithmetic series involves:

• Identifying the first term and the last term in the sequence, along with the number of terms.
• Use the formula for arithmetic series, which is:\[ S_n = \frac{n}{2} \, (a + l) \]Where \( S_n \) is the sum of the series, \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

This formula helps sum up an arithmetic series quickly. In our holiday song example, using this formula gives the total number of gifts as 78, when summed up for the 12 days.

sum of natural numbers

The sum of natural numbers is a simple yet fundamental mathematical operation. Natural numbers are all positive integers starting from 1.

When you need to sum the first \( n \) natural numbers, there is a direct formula used:\[ S_n = \frac{n(n+1)}{2} \]This formula derives from the arithmetic series properties and is particularly useful for quickly finding the total of sequential numbers.

In our exercise regarding the "Twelve Days of Christmas," this formula shows how many gifts are given in total by the 12th day. By substituting \( n = 12 \), you can compute:

• Plug \( n = 12 \) into the formula: \[ S_{12} = \frac{12(12+1)}{2} \]
• Calculate: \( \frac{12 \times 13}{2} = 78 \)

Hence, 78 gifts are given after twelve days, illustrating the power and efficiency of summing natural numbers using the formula. This simple calculation can be applied broadly in various numerical and practical situations.