Problem 38
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
Verified Answer
(a) 364 gifts. (b) \( T_n = \frac{1}{6}n(n+1)(n+2) \).
1Step 1: Understand the Gift Pattern
In the song, each day builds upon the previous by adding a new layer of gifts. On the first day, 1 gift is given. On the second day, 1 + 2 gifts, on the third day, 1 + 2 + 3 gifts, and so on. Thus, the gifts given on day n is the sum of the first n natural numbers.
2Step 2: Use the Sum Formula
The sum of the first n natural numbers is given by the formula \( S = \frac{n(n + 1)}{2} \). On each day, we calculate the cumulative sum of these gifts.
3Step 3: Calculate the Total Number of Gifts for 12 Days
Calculate the total number of gifts for each day and then sum them up:- Day 1: \( \frac{1 \times 2}{2} = 1 \)- Day 2: \( \frac{2 \times 3}{2} = 3 \)- Day 3: \( \frac{3 \times 4}{2} = 6 \)- ... - Day 12: \( \frac{12 \times 13}{2} = 78 \)Add up all the gifts: \( 1 + 3 + 6 + ... + 78 \).
4Step 4: Total Sum Calculation
To find the total, use the formula for the sum of consecutive natural numbers for each day and sum them together from Day 1 to Day 12:\[ \text{Total gifts} = \sum_{k=1}^{12} \frac{k(k+1)}{2} \]Calculating directly or leveraging the properties of sequences eventually gives:\[ \text{Total gifts} = 364 \]
5Step 5: Derive the Formula for General n Days
We know each day's gifts are \( \frac{k(k+1)}{2} \). The total gifts over n days can be obtained by:\[ T_n = \sum_{k=1}^{n} \frac{k(k+1)}{2} \]This means we find the sum of each day's total gifts, which simplifies to: \[ T_n = \frac{1}{6}n(n+1)(n+2) \] by calculating and summing up the cumulative nature.
Key Concepts
Sum of Natural NumbersCumulative SumsChristmas Gift Problem
Sum of Natural Numbers
The sum of the first few natural numbers is a concept frequently encountered in mathematics. It's illustrated well by the exercise related to the "Twelve Days of Christmas" song.
Each day, more gifts build upon the previous days, following a pattern. On day one, there is 1 gift, on day two, the sum becomes 1 + 2, on day three it's 1 + 2 + 3, and so forth. Mathematically, this sequence becomes known as the sum of the first n natural numbers.
The formula to calculate such a sum is efficiently given by \( S = \frac{n(n + 1)}{2} \). This formula generates the sum of all numbers from 1 to n, fundamental to solving the problem of counting gifts over the twelve days.
Each day, more gifts build upon the previous days, following a pattern. On day one, there is 1 gift, on day two, the sum becomes 1 + 2, on day three it's 1 + 2 + 3, and so forth. Mathematically, this sequence becomes known as the sum of the first n natural numbers.
The formula to calculate such a sum is efficiently given by \( S = \frac{n(n + 1)}{2} \). This formula generates the sum of all numbers from 1 to n, fundamental to solving the problem of counting gifts over the twelve days.
Cumulative Sums
Cumulative sums refer to the running total of adding up elements sequentially from a series or sequence. In simpler terms, you progressively add each element's value to the total sum as you move through a list.
For the Twelve Days of Christmas problem, each day's gifts are calculated as the sum of all previous days plus that day's addition. For instance, on the second day, you add 2 gifts to the previous total (which was 1).
Understanding cumulative sums is helpful in various scenarios, such as calculating total sales over a period or finding the aggregate value of several investments. In this exercise, the concept helps in computing the total number of gifts provided over the entire twelve days.
For the Twelve Days of Christmas problem, each day's gifts are calculated as the sum of all previous days plus that day's addition. For instance, on the second day, you add 2 gifts to the previous total (which was 1).
Understanding cumulative sums is helpful in various scenarios, such as calculating total sales over a period or finding the aggregate value of several investments. In this exercise, the concept helps in computing the total number of gifts provided over the entire twelve days.
Christmas Gift Problem
The Christmas gift problem is a whimsical yet instructive exercise that teaches students about sequences and sums in a fun context. It uses the narrative of a song where gifts are given in increasing quantities each day over twelve days.
By examining this problem, students learn how to apply formulas for sums in practical scenarios. The task of finding the total number of gifts combines several key mathematical concepts, making it an excellent learning tool.
It not only involves calculating the gifts for each day but also requires understanding the cumulative nature of these numbers. Through simplifying expressions and solving sums, students derive a general formula for any number of days, which demonstrates how standardized mathematics can solve practical and creative problems.
By examining this problem, students learn how to apply formulas for sums in practical scenarios. The task of finding the total number of gifts combines several key mathematical concepts, making it an excellent learning tool.
It not only involves calculating the gifts for each day but also requires understanding the cumulative nature of these numbers. Through simplifying expressions and solving sums, students derive a general formula for any number of days, which demonstrates how standardized mathematics can solve practical and creative problems.
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