Problem 9 - 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. \(n^{2}+n\) is divisible by 2

Step-by-Step Solution

Answer

The number of gifts sent on the nth day of Christmas follows the pattern of the sum of the first n integers, given by the formula \(\frac{n(n+1)}{2}\). We prove that this formula is always divisible by 2, as either n or \(n+1\) must always be even, thus ensuring that the number of gifts sent on each day will be an integer value.

Step 1: Understand the Arithmetic Series Formula

The formula for the sum of the first n integers is derived from the arithmetic series formula. An arithmetic series is a series whose terms are in an arithmetic progression, which means the difference between any two consecutive terms is always the same. The sum of an arithmetic series can be calculated using the formula: \(S_n = \frac{n(a_1 + a_n)}{2}\) Where \(S_n\) is the sum of the series, n is the number of terms, \(a_1\) is the first term, and \(a_n\) is the nth term of the series. In this case, \(a_1 = 1\) and we can find the nth term as \(a_n = n\) because the difference between consecutive terms is 1.

Step 2: Find the Sum of the First n Integers

Using the arithmetic series formula, we can find the sum of the first n integers: \(S_n = \frac{n(1 + n)}{2}\) Or, re-written as: \(S_n = \frac{n(n+1)}{2}\) This is the formula for the number of gifts sent on the nth day of Christmas.

Step 3: Prove the Sum Formula is Divisible by 2

To prove that the sum formula is divisible by 2 or that it always results in an integer value, we need to show that either n or \(n+1\) is an even number, as the product of an even number and any other number is always even. Take two consecutive numbers, one being n, and the other being \(n+1\). In this pair, one number must be even because the difference between even and odd numbers in a sequence is always 1. So, either n or \(n+1\) must be even.

Step 4: Conclusion

We have found that the number of gifts sent on the nth day of Christmas follows the pattern of the sum of the first n integers. We also proved that the formula \(\frac{n(n+1)}{2}\) is always divisible by 2 by showing that either n or \(n+1\) is always even. This means that the number of gifts sent on each of the "Twelve Days of Christmas" will be an integer value.

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is created by adding a constant difference to the previous term. This constant is called the "common difference." An example of an arithmetic progression is the sequence 1, 2, 3, 4, 5, where the common difference is 1.
For any arithmetic progression, you can describe the n-th term using the formula:

• The first term of the sequence is denoted by \(a_1\), which in our example is 1.
• The common difference \(d\), is the amount added each time (here, it's also 1).
• The n-th term \(a_n\) is found using: \(a_n = a_1 + (n-1) \, d\).

Thus, in an arithmetic progression, everything centers on this regular interval, making calculations predictable. This regularity allows us to efficiently find terms and sums in the sequence.

Sum Formula

The sum formula for an arithmetic series allows us to quickly calculate the sum of its terms. It is especially helpful to find how many gifts are sent on each of the Twelve Days of Christmas. For an arithmetic sequence with a first term \(a_1\) and a final n-th term \(a_n\), the sum \(S_n\) can be found using:

• \(S_n = \frac{n(a_1 + a_n)}{2}\)

In our Christmas example with the sequence of gifts, each day corresponds to the sum of the first n natural numbers. Here, \(a_1\) is 1 and \(a_n\) is n.
This simplifies to \(S_n = \frac{n(n+1)}{2}\), which directly calculates the total number of gifts given by the nth day. Essentially, this sum formula streamlines finding cumulative totals, transforming what could be lengthy calculations into simple multiplication and division.

Divisibility Proof

The solution's divisibility proof shows how the sum from the previous formula \(\frac{n(n+1)}{2}\) is always an integer. This involves showing that for any integer n, this quantity is divisible by 2.
Consider the two consecutive numbers, n and n+1:

• In any sequence of two consecutive numbers, one number is always even.
• The product of an even number (either n or n+1) with any other number gives an even result.
• This means that \(n(n+1)\) is always even, as it involves multiplying either n or n+1, which is even.

Therefore, when you divide \(n(n+1)\) by 2, the quotient is a whole number. This confirms the formula's reliability in determining how many gifts are given each Christmas day while preserving the integrity of the values being perfect integers. This logic showcases a clever use of arithmetic properties to enhance understanding of sequences and series.