Problem 13 - Discrete Mathematics with Applications

Question

Suppose that on the first day of Christmas you sent your love 1 gift, \(1+2\) gifts on the second day, \(1+2+3\) gifts on the third day, and so on. Find the number of gifts sent on the 12 th day.

Step-by-Step Solution

Answer

On the 12th day of Christmas, you sent your love 78 gifts.

Step 1: Identify the pattern

The pattern of gifts sent is as follows: 1st day: 1 2nd day: 1 + 2 3rd day: 1 + 2 + 3 ... 12th day: 1 + 2 + 3 + ... + 12 Notice that the pattern is the sum of the first n consecutive integers, where n corresponds to the day number.

Step 2: Find the formula for triangular numbers

The sum of the first n natural numbers is called a triangular number, and it can be calculated using the formula: \[T_n = \frac{n(n+1)}{2}\] where \(T_n\) is the nth triangular number and n is the number of terms (or the day number in this case).

Step 3: Calculate the number of gifts sent on the 12th day

To find the number of gifts sent on the 12th day, we will plug n = 12 into the formula: \[T_{12} = \frac{12(12+1)}{2}\]

Step 4: Solve the equation

Now we just need to solve for \(T_{12}\): \[T_{12} = \frac{12\times13}{2}\] \[T_{12} = \frac{156}{2}\] \[T_{12} = 78\]

Step 5: State the answer

On the 12th day of Christmas, you sent your love 78 gifts.

Key Concepts

Sum of Consecutive Integers

In this exercise, we delve into the concept of the sum of consecutive integers, which is a foundational topic in mathematics. You might consider this concept as one way to count collections of things, which appears often in everyday life. For example, if you are walking up a flight of stairs and you count each step, you are summing consecutive integers.

The sum of consecutive integers can be understood as adding numbers in a sequence without skipping any number in between. Suppose we want to find the sum of the first "n" integers:

• Start from 1, then add 2, followed by adding 3, and continue this way till "n".
• The challenge lies in finding a quick method to calculate this sum without adding each number individually.

In our exercise, the sum for each day reflects this pattern, which escalates with each passing day.

The calculated result for each of the summed integers on a particular day gives us what we call a triangular number, which plays a crucial role in understanding how numbers accumulate through different days as shown in the original example of gifts being sent.

Discrete Mathematics

Discrete mathematics encompasses a set of mathematical topics that deal with discrete objects, or in simpler terms, countable things. It’s the backbone for much of computer science, game theory, and network models. Within this field, one of the central topics is the analysis of patterns and structures which exist distinctly or separately.

In the context of our exercise, calculating the sum of gifts over several days using the formula for triangular numbers is a task that can be reasoned through discrete mathematics.

• Each step, each gift, and each day are individual entities that can be broken down logically and managed separately, not unlike how discrete numbers function.
• The very pattern of gifts sent in increasing sums day by day forms a discernible sequence, emphasizing the principles of discrete mathematics where finite or countable collections are studied.

This shows how findings in this domain are applicable in understanding everyday numerical patterns and processes.

Mathematical Patterns

Mathematical patterns involve sequences and regularities within numbers that enable us to predict further elements in a given sequence. They help unlock the underlying order behind a set of numbers, often leading to intriguing mathematical discoveries and applications.

Triangular numbers, such as those seen in this exercise, illustrate a mathematical pattern. Let's understand this:

• By noticing the pattern of sums for successive days (1, 1+2, 1+2+3, etc.), we see an increasing sequence which can be described by a clear formula.
• Using the formula \[T_n = \frac{n(n+1)}{2}\]allows us to derive the triangular number for any day, thus simplifying the effort needed to calculate the result each day lends to this mathematical pattern.

Recognizing these patterns helps students not only solve individual problems but develop a critical understanding of how structured sequences arise and are calculated. From an educational standpoint, it is about developing a mathematical lens through which various phenomena can be explored and understood.