Problem 38

Question

In the song The Twelve Days of Christmas, my true love gave me 1 gift on the first day, $1+2$ gifts on the second day, $1+2+3$ gifts on the third day, and so on for 12 days. (a) Find the total number of gifts given in 12 days. (b) Find a simple formula for $T_{n}$, the total number of gifts given during a Christmas of $n$ days.

Step-by-Step Solution

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Answer

(a) 364 gifts in 12 days. (b) Formula: $T_n = \frac{n(n+1)(n+2)}{6}$.

1Step 1: Understand the Pattern

The gifts given each day follow a cumulative sequence from the first to the twelfth day. Each day's gifts are the sum of all natural numbers from 1 to the day number.

2Step 2: Define the Formula for the nth Day

For any day $n$, the number of gifts given on that day is the sum of the first $n$ natural numbers, which can be calculated using the formula $S_n = \frac{n(n+1)}{2}$.

3Step 3: Calculate Daily Gifts for 12 Days

Using the formula $S_n = \frac{n(n+1)}{2}$, compute the number of gifts given each day: These are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, and 78 gifts from day 1 to day 12.

4Step 4: Sum All Gifts for 12 Days

Add the results from each of the 12 days: $1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78$. The total number of gifts over 12 days is 364.

5Step 5: Find the Formula for Total Gifts over n Days

The total gifts $T_n$ over $n$ days is essentially the sum of triangular numbers up to $n$. The formula for the sum of the first $n$ triangular numbers is $T_n = \frac{n(n+1)(n+2)}{6}$.

Key Concepts

Cumulative SequenceTriangular NumbersSum of Natural Numbers

Cumulative Sequence

A cumulative sequence is where each term in the sequence is the sum of all the previous terms plus the current term. It's like collecting all the gifts received over a period, where each day builds on the previous. In the Twelve Days of Christmas problem, this sequence starts with one gift on the first day, and the number of gifts increases each day by the sum of the next number in line. This pattern helps depict how quantities increase over time. It’s crucial for problems involving summation, as understanding the accumulation allows the calculation of a total or end value.
To illustrate a cumulative sequence:

On Day 1: 1 gift.
On Day 2: 1 (from day 1) + 2 = 3 gifts.
On Day 3: 1 + 2 (from previous days) + 3 = 6 gifts.

As each day passes, you are not just adding another gift; you're adding more than one because you collect all previous days' gifts plus the current day's new addition. This cumulative nature makes the sequence interesting and highlights the acceleration of total gifts received over time.

Triangular Numbers

Triangular numbers are a special type of number that can form an equilateral triangle when represented as dots. These numbers appear prominently when summing a sequence of consecutive natural numbers. A triangular number is simply the total number of dots needed to complete the triangle, and it's calculated using the formula: $T_n = \frac{n(n+1)}{2}$.
For example, to find the 3rd triangular number:

Substitute $n = 3$ in the formula: $T_3 = \frac{3(3+1)}{2} = 6$.

This means the total number of gifts, when using triangular numbers for each day, grows exponentially as more days pass. The importance of these numbers lies in their ability to simplify calculations involving sequential summing, as shown in exercises like the Twelve Days of Christmas where gifts are given in proportions similar to these numbers.

Sum of Natural Numbers

The sum of natural numbers from 1 to any number $n$ is pivotal in various mathematical calculations. It's essential in adding the simple, sequential counting numbers starting from 1. The formula to easily calculate the sum is given by: $S_n = \frac{n(n+1)}{2}$, and this formula efficiently provides the needed totals without needing to manually add each number.
Applying this concept:

For $n = 4$, sum the numbers 1 to 4: $S_4 = \frac{4(4+1)}{2} = 10$.

Understanding the sum of natural numbers is critical for recognizing patterns in sequences and solving problems where accumulation is involved, like calculating the total number of gifts in cumulative sequences. It provides a powerful shortcut to what could otherwise be a tedious process of summation. Thus, using this formula, one can quickly determine how much has accumulated each day, as shown in problem-solving scenarios like gift-giving sequences.

Problem 37

Problem 38

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