Problem 13 - Discrete Mathematics with Applications

Question

(Twelve Days of Christmas) 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 a total of 78 gifts to your love.

Step 1: Set up the Formula for the Sum of First N Integers

We will use the formula for the sum of the first N positive integers, which is given by \(S_n = \frac{n(n+1)}{2}\) where n is the number of terms (or integers) we want to sum.

Step 2: Substitute N with 12

We will substitute n with 12 to find the number of gifts sent on the 12th day. \(S_{12} = \frac{12(12+1)}{2}\)

Step 3: Simplify the Expression

Now we will simplify the expression to find the total number of gifts: \(S_{12} = \frac{12(13)}{2}\)

Step 4: Calculate the Number of Gifts on the 12th Day

Multiply the numbers and divide by 2 as per the formula: \(S_{12} = \frac{156}{2}\) \(S_{12} = 78\) So, on the 12th day of Christmas, you sent a total of 78 gifts to your love.

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference of any two successive members is a constant called the common difference, denoted as 'd'. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2.

When discussing the Twelve Days of Christmas and the total gifts sent each day, we're essentially looking at the sum of an arithmetic series where each day represents a term in the AP with a common difference of 1. On the first day, the number of gifts is 1, on the second day it is 1+2, on the third day it is 1+2+3, and so on. Each day's gifts are an accumulated sum of the days before it plus that day's number, forming an arithmetic sequence of sums.

Finding the n-th Term of an AP

For a given AP, the n-th term can be found using the formula: \[ a_n = a_1 + (n - 1)d \]where \( a_1 \) is the first term and 'n' is the number of terms. The sequence of total gifts each day is a series of such n-th terms.

Series Summation

In mathematics, series summation is the addition of a sequence of any kind of numbers, called the terms of the series. Specifically, when we discuss the sum of the first N positive integers, we are talking about the series summation of an arithmetic progression.

The formula used in the Christmas gift example, \[ S_n = \frac{n(n+1)}{2} \]is derived from summing the first N terms of an AP when the common difference is 1. This formula is a special case and applies to the sequence of 1, 2, 3, ..., N. It is a crucial concept for understanding the number of gifts sent on the 12th day, as the total number of gifts each day is a series summation up to that day.

Visualizing Series Summation

One method to visualize this summation is to imagine a triangle with a base and height of 'n', where each level of the triangle represents the sum of integers up to that point. The area of the triangle reflects the total sum we are trying to compute, thus offering a geometric interpretation of series summation.

Mathematical Induction

Mathematical induction is a method of mathematical proof typically used to prove that a given statement is true for all natural numbers. It consists of two parts: the base case and the inductive step.

For the Twelve Days of Christmas problem, we could use mathematical induction to prove the sum formula works for any 'n' number of days. Here's how we could apply it:

Base Case

The base case would verify the formula works for the first natural number, usually 1. Here, we'd show that for \( n = 1 \), the formula yields the right number of gifts, which is 1 in this case.

Inductive Step

Next, we assume the formula works for an arbitrary positive integer 'k' and then prove that if it holds for 'k', it must also hold for 'k+1'. This completes the inductive step and confirms the general case, thereby proving the summation formula is indeed valid for all days of this festive tradition.