Question

An English Christmas carol, "The Twelve Days of Christmas, " dates from the late \(1700 \mathrm{~s}\). The 12 verses in the song are cumulative, each verse adding an additional gift given by "my true love." The twelfth verse says "On the twelfth day of Christmas, my true love gave to me ..." 12 Drummers Drumming 11 Pipers Piping 10 Lords-a-Leaping \(\ldots\) and so forth down to ... 3 French Hens 2 Turtle Doves And a Partridge in a Pear Tree. a. Use Gauss's formula to find the total number of gifts given on Day 12 . b. How many total gifts are given over all 12 days? Hint: $$ 1(2)+2(3)+3(4)+\ldots+n(n+1)=\frac{n(n+1)(n+2)}{3} $$

Short answer

78 gifts on Day 12; 364 gifts over all 12 days.

Step-by-step solution

Understand Gauss's formula

Gauss's formula for the sum of the first \( n \) natural numbers is \( \frac{n(n+1)}{2} \). We can use this to find the total number of gifts given on Day 12.

Apply Gauss's formula to Day 12

For Day 12, calculate the sum of numbers from 1 to 12. Using Gauss's formula: \[ \text{Sum} = \frac{12 \cdot (12 + 1)}{2} = \frac{12 \cdot 13}{2} = 78 \]. Thus, 78 different gifts are given on Day 12.

Understand the cumulative gift pattern for all 12 days

Every day accumulates its specific number of gifts given plus all the gifts from the previous days. For Day 1, 1 gift; Day 2, 1+2 gifts; Day 3, 1+2+3 gifts, and so on up to Day 12.

Use the hint formula to find total gifts over 12 days

The given hint formula is \[ 1(2) + 2(3) + 3(4) + \ldots + n(n+1) = \frac{n(n+1)(n+2)}{3} \]. Substituting \( n = 12 \):\[ \text{Total gifts} = \frac{12 \cdot 13 \cdot 14}{3} = 364 \]. Thus, a total of 364 gifts are given over all 12 days.

Key concepts

Gauss's Formula

Gauss's Formula is a clever technique often used in mathematics to simplify the process of adding consecutive numbers. It is named after the famous German mathematician Carl Friedrich Gauss. The formula calculates the sum of the first \( n \) natural numbers and is given by \( \frac{n(n+1)}{2} \). Let's break it down: consider the task of summing the numbers from 1 to 12. Instead of adding each number one by one, Gauss's Formula provides a quick computation. In essence:

• Multiply the largest number in the series \( n \) by the next consecutive number \( n+1 \).
• Divide the product by 2.

Following these steps for 12 days, with \( n = 12 \), gives \( \frac{12 \cdot 13}{2} = 78 \), indicating 78 gifts were given on the twelfth day. This method is valuable because it reduces the effort required for calculating sums and ensures accuracy.

Summation Techniques

Summation techniques involve strategies and formulas that help efficiently calculate the sum of sequences or series of numbers. They are crucial in mathematics education as they streamline otherwise complicated processes. In the context of "The Twelve Days of Christmas," the challenge is to determine the total number of gifts given over 12 days, using a series pattern. The provided hint formula \( 1(2) + 2(3) + 3(4) + \ldots + n(n+1) = \frac{n(n+1)(n+2)}{3} \) is a specialized formula for such cumulative problems.Here's how it works:

• This sums the product of each day index \( n \) and the gifts on that day \( n+1 \), reflecting both the day's position and its incremental nature.
• The solution involves substituting the total days \( n = 12 \) into the formula, resulting in \( \frac{12 \cdot 13 \cdot 14}{3} = 364 \).

By using summation techniques, one can handle adding sequences more confidently, speeding up calculations, and understanding more about how numbers interact in series.

Problem Solving Strategies

Problem-solving strategies in mathematics are approaches or techniques that help tackle a problem efficiently and effectively. With exercises like "The Twelve Days of Christmas," having a strategy simplifies the process. Effective strategies often include:

• Understanding the Problem: Carefully read the problem to grasp what is being asked. Like identifying that the exercise is about adding gifts for specific days.
• Choosing the Right Tools: Selecting the appropriate formulas, like Gauss's or the summation hint, to make calculations efficient and accurate.
• Breaking Down the Problem: Divide it into manageable parts; here, calculate Day 12's gifts first, then total over 12 days.
• Checking Work: Verify results with logical reasoning or alternative methods if possible.

Applying these steps enhances understanding and contributes to mathematical confidence, making it easier to solve similar problems in the future.