Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

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

Answer

(a) 364 gifts were given in 12 days. (b) The formula for total gifts for \( n \) days is \( T_n = \frac{n(n+1)(n+2)}{6} \).

Step 1: Understanding the Problem

In the song, the true love gives gifts in increasing amounts each day. On day 1, 1 gift is given; on day 2, a total of 3 gifts (1+2); on day 3, 6 gifts (1+2+3), continuing up to day 12.

Step 2: Identifying the Pattern

We observe that each day's gift follows the sum of the first n natural numbers for that day. For instance, on day 3, it's 1+2+3 = 6 gifts, which is a sum of the first 3 natural numbers.

Step 3: Using Arithmetic Series Formula

The sum of the first n natural numbers is given by the formula: \( S_{n} = \frac{n(n+1)}{2} \). This formula can help calculate total gifts given cumulatively up to any specific day.

Step 4: Calculating the Total for 12 Days

Using the formula, we need the sum of these sums from 1 to 12: \( T = S_1 + S_2 + \ldots + S_{12} \). Substitute \( S_n = \frac{n(n+1)}{2} \) for each day and sum them.

Step 5: Summing the Sums Simplified

To simplify: \( T = \sum_{n=1}^{12} \frac{n(n+1)}{2} \). Simplify further, this becomes \( T = \frac{1}{2} \sum_{n=1}^{12} n(n+1) \). This breaks to \( \frac{1}{2} (1^2 + 2^2 + \ldots + 12^2 + 1 + 2 + \ldots + 12) \).

Step 6: Calculating Each Part

Calculating the numerical components: \( \sum_{n=1}^{12} n = \frac{12(12+1)}{2} = 78 \) and \( \sum_{n=1}^{12} n^2 = \frac{12(12+1)(2*12+1)}{6} = 650 \).

Step 7: Final Total Calculation

Inserting sums back gives \( T = \frac{1}{2} (650 + 78) = \frac{1}{2} (728) = 364 \). So 364 gifts were given in 12 days.

Step 8: Finding General Formula for \( T_n \)

The approach to find \( T_n \) involves recognizing that \( T_n = \sum_{k=1}^{n} \frac{k(k+1)}{2} \). This simplification lands on a cubic formula resulting in \( T_n = \frac{n(n+1)(n+2)}{6} \).

Key Concepts

Sum of Natural Numbers

The sum of natural numbers is a fundamental idea in mathematics. Natural numbers are the numbers you naturally count with, starting from 1, 2, 3, and so on.
Understanding how to sum them is key in various scenarios, like our Christmas gift problem. For any natural number sequence up to a number \( n \), this can be calculated using a simple formula:

• Sum Formula: \( S_n = \frac{n(n+1)}{2} \)

This formula effectively adds up all numbers from 1 to \( n \) by pairing numbers from each end moving inward (e.g., 1 with 12, 2 with 11 if considering up to 12), a handy trick first formalized by Carl Friedrich Gauss.Using this formula saves time and hassle compared to adding up each number one by one.

Arithmetic Series Formula

An arithmetic series is a series of numbers where each term increases by a constant amount. This concept is different from an arithmetic sequence but related.
The formula for the sum of an arithmetic series gives us an efficient way to sum such sequences:

• Series Formula: \( S_n = \frac{n}{2} (a + l) \)
• Increments: Each step in the arithmetic series increases by the same amount.

In our gift example, each day's gifts form part of an arithmetic series. By plugging the values, you can efficiently find the total gift count for any number of days defined by this logical pattern.

Mathematical Patterns

Spotting mathematical patterns helps in predicting outcomes and solving complex problems efficiently.
In our scenario, recognizing that the daily gifts follow a set pattern (sum of natural numbers) lets us use the series formula directly. For instance, each day follows the pattern:

• 1st day: 1 gift
• 2nd day: 1+2 gifts
• 3rd day: 1+2+3 gifts, etc.

These sums themselves form a new sequence, which can be summed using simple techniques, aiding in understanding and forming a general solution.

Series Summation Techniques

Summation techniques are tools used to find the sum of a sequence of numbers without direct computation number-by-number. One efficient technique is transforming the problem into a sum of the series formula:

• Break down the sum (like gifts each day) into known smaller sums.
• Use known summation formulas to simplify and calculate quickly.

For our exercise, using the formula \( T_n = \frac{n(n+1)(n+2)}{6} \) is a result of such a technique. This allows us to find the total number of gifts given for any series of consecutive days, providing a powerful shortcut compared to calculating manually.