Problem 6 - Discrete Mathematics with Applications

Question

(Twelve Days of Christmas) Suppose you sent your love 1 gift on the first day of Christmas, \(1+2\) gifts on the second day, \(1+2+3\) gifts on the third day and so on. $$ \sum_{i=1}^{n} i^{2}=\frac{(n+1)(2 n+1)}{6} $$

Step-by-Step Solution

Answer

The total number of gifts sent till the nth day of Christmas can be found using the formula: Total Gifts = \(\frac{1}{4} \sum_{i=1}^{n} i^2 (i+1)^2\) Simply substitute the desired value of n into the formula to determine the total number of gifts sent.

Step 1: Determine the number of gifts sent on each day

We will first write down the pattern of gifts sent on each day: 1st day: 1 gift 2nd day: 1 + 2 = 3 gifts 3rd day: 1 + 2 + 3 = 6 gifts 4th day: 1 + 2 + 3 + 4 = 10 gifts and so on.

Step 2: Notice the triangular numbers

We can observe that the number of gifts sent on each day is a sequence of triangular numbers: \(1, 3, 6, 10, \ldots \) A triangular number \(T_n\) is the sum of the first n natural numbers, which can be written as: \(T_n = \sum_{i=1}^{n} i\) for n = 1, 2, 3,...

Step 3: Express the problem as a summation of squares

Since we want to find the total number of gifts sent till the nth day of Christmas, we need to express the problem as a sum of the triangular numbers: \(\sum_{i=1}^{n} T_{i}^2\) Here, \(\sum_{i=1}^{n} T_{i}^2\) gives the total number of gifts sent till the nth day.

Step 4: Apply the summation formula for the sum of squares

Using the sum of squares formula - \(\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\), we can now find the total number of gifts sent: \(\sum_{i=1}^{n} T_{i}^2 = \sum_{i=1}^{n} (\frac{i(i + 1)}{2})^2\) Here, we replaced the triangular numbers with their formula, which is \(\frac{i(i + 1)}{2}\).

Step 5: Simplify the expression

Let's simplify the expression to find the total gifts sent: \(\sum_{i=1}^{n} (\frac{i(i + 1)}{2})^2 = \frac{1}{4} \sum_{i=1}^{n} i^2 (i+1)^2\)

Step 6: Final expression

The final expression for the total number of gifts sent till the nth day of Christmas can be given as: Total Gifts = \(\frac{1}{4} \sum_{i=1}^{n} i^2 (i+1)^2\) Now, you can find the total number of gifts sent after any nth day of Christmas by plugging in the value of n in the formula.

Key Concepts

Triangular Numbers

Triangular numbers are a special kind of number that represents the pattern of dots that form an equilateral triangle. In the context of the "Twelve Days of Christmas" problem, they help illustrate the increasing number of gifts given each day. The first few triangular numbers are 1, 3, 6, 10, and so on.

• The nth triangular number, denoted as \(T_n\), is calculated as the sum of the first n natural numbers.
• Mathematically, it is represented by the formula: \(T_n = \sum_{i=1}^{n} i = \frac{n(n+1)}{2}\).
• This pattern showcases how the number of dots increases to complete each subsequent triangular number.

For example, the 3rd triangular number is calculated by adding 1 + 2 + 3, which equals 6. This formula can be applied to find any triangular number, simplifying the process of dealing with large series.

Sum of Squares Formula

Understanding the sum of squares formula is crucial when dealing with problems that involve summing more complex series, such as in the "Twelve Days of Christmas." The sum of squares involves adding the squares of the first n natural numbers and is given by:

• Formula: \( \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \)
• This formula provides a quick method to calculate the sum without adding each number manually.
• It is particularly helpful in exercises where efficiency and accuracy are needed.

Applying this formula to each triangular number squared helps establish the overall total gifts sent. For instance, using this approach transforms a seemingly daunting task into a manageable computation process.

Arithmetic Series

An arithmetic series refers to the sum of the terms in an arithmetic sequence, where each term increases by a constant amount. This is relevant to understanding the regularly increasing number of gifts in the "Twelve Days of Christmas."

• An arithmetic sequence is defined by its first term and its common difference between terms.
• The sum of an arithmetic series can be found using the formula: \( S_n = \frac{n}{2} (a_1 + a_n) \), where \(a_1\) is the first term and \(a_n\) is the nth term in the sequence.
• This formula gives us a straightforward way to find the sum of any given series in less time.

When calculating the total gifts over multiple days, recognizing the use of arithmetic series simplifies how we sum the pattern of gifts provided each day. By grouping related numbers and applying these formulas efficiently, students can solve complex summation problems more easily.