Problem 5
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. $$ \sum_{i=1}^{n}(2 i-1)=n^{2} $$
Step-by-Step Solution
Verified Answer
To find the total number of gifts sent to your love during n days of Christmas, follow the given pattern and use the formula \(\sum_{i=1}^{n}(2 i-1)=n^2\). Just plug in the value of n in the formula, and you get the total number of gifts sent during n days.
1Step 1: Identify the problem
The problem asks us to calculate the total number of gifts sent to your love during n days of Christmas, following the given pattern. We will be using the provided formula \(\sum_{i=1}^{n}(2 i-1)=n^2\) to find the answer.
2Step 2: Understanding the formula
The given formula involves a summation, \(\sum_{i=1}^{n}(2 i-1)\), which calculates the number of gifts sent on each day, where i is the number of the current day and n is the total number of days. The formula states that the sum of all the gifts sent during n days is equal to \(n^2\).
Example: For the first day, the total gifts = \(2(1)-1 =1\), for the second day, the gifts will be = \(2(2)-1 = 3\), so on the second day, you sent your love \(1+3=4\) gifts.
3Step 3: Plug n in the formula
Now, we just need to plug in the value of n in the formula to find the total number of gifts sent during n days. For instance, if n=3 (3 days of Christmas), then:
Total gifts = \(n^2 = 3^2 = 9\)
So, during 3 days of Christmas, you sent your love 9 gifts following the given pattern.
4Step 4: Conclusion
To find the total number of gifts sent to your love during n days of Christmas, follow the given pattern and use the formula \(\sum_{i=1}^{n}(2 i-1)=n^2\). Just plug in the value of n in the formula, and you get the total number of gifts sent during n days.
Key Concepts
Arithmetic SequencesSeries and SummationMathematical Induction
Arithmetic Sequences
An arithmetic sequence is a list of numbers where each term is obtained by adding a fixed number to the previous term. This fixed number is known as the common difference. In our Christmas gift-giving scenario, we notice the number of gifts sent each day doesn't follow a simple arithmetic sequence, but rather a cumulative total based on increasing arithmetic sequences.
For example, on the first day, just 1 gift is sent. On the second day, the sequence of gifts is 1 and 3. While 1 and 3 are not consecutive terms in an arithmetic sequence, their sum is part of an increasing series of numbers that follow the pattern of adding an additional term from an arithmetic sequence characterized by a common difference of 2.
It is this pattern of accumulation that leads to the overarching formula provided in the exercise, which can be derived by recognizing the underlying arithmetic sequence and summing up its terms for each day.
For example, on the first day, just 1 gift is sent. On the second day, the sequence of gifts is 1 and 3. While 1 and 3 are not consecutive terms in an arithmetic sequence, their sum is part of an increasing series of numbers that follow the pattern of adding an additional term from an arithmetic sequence characterized by a common difference of 2.
It is this pattern of accumulation that leads to the overarching formula provided in the exercise, which can be derived by recognizing the underlying arithmetic sequence and summing up its terms for each day.
Series and Summation
A series in mathematics is the sum of the terms of a sequence, and summation is the process of adding these numbers together. We often use the summation symbol \( \Sigma \), which is the Greek capital letter sigma, to signify the sum of a sequence. In the context of the Christmas problem, summation helps us find the total number of gifts given over the course of n days.
The formula provided, \( \sum_{i=1}^{n}(2i-1) = n^2 \), is a concise way of stating that when you add up all the odd numbers from 1 to an odd number designated by 2n-1, the result will be the square of n. This kind of series is particularly interesting because it demonstrates a pattern that isn't initially obvious — that the sum of the first n odd numbers is always a perfect square, a neat characteristic of these numbers.
To improve the understanding of series and summation, one can practice by manually adding the first few terms to see how they add up to the square of the count of terms. Recognizing these patterns aids in grasping the concept of series and the efficiency of mathematical notation in describing them.
The formula provided, \( \sum_{i=1}^{n}(2i-1) = n^2 \), is a concise way of stating that when you add up all the odd numbers from 1 to an odd number designated by 2n-1, the result will be the square of n. This kind of series is particularly interesting because it demonstrates a pattern that isn't initially obvious — that the sum of the first n odd numbers is always a perfect square, a neat characteristic of these numbers.
To improve the understanding of series and summation, one can practice by manually adding the first few terms to see how they add up to the square of the count of terms. Recognizing these patterns aids in grasping the concept of series and the efficiency of mathematical notation in describing them.
Mathematical Induction
Mathematical induction is a powerful proof technique used to show that a statement or formula is true for all natural numbers. The process starts with checking if the statement is true for an initial value, often done for the number 1, known as the base case.
After establishing the base case, we assume that the statement holds for some arbitrary natural number k, and then show that if it holds for k, it must also hold for k+1. This step is called the inductive step. The principle of mathematical induction stems from the domino effect; if you can show that the first domino falls (base case) and every domino will knock down the subsequent one (inductive step), then all dominoes will fall.
Using mathematical induction, we could prove the given formula \( \sum_{i=1}^{n}(2i-1) = n^2 \) by showing it's true for n=1 and then assuming it's true for n=k and showing it must be true for n=k+1. This logical progression solidifies our understanding of the formula and ensures that it holds for all natural numbers n.
After establishing the base case, we assume that the statement holds for some arbitrary natural number k, and then show that if it holds for k, it must also hold for k+1. This step is called the inductive step. The principle of mathematical induction stems from the domino effect; if you can show that the first domino falls (base case) and every domino will knock down the subsequent one (inductive step), then all dominoes will fall.
Using mathematical induction, we could prove the given formula \( \sum_{i=1}^{n}(2i-1) = n^2 \) by showing it's true for n=1 and then assuming it's true for n=k and showing it must be true for n=k+1. This logical progression solidifies our understanding of the formula and ensures that it holds for all natural numbers n.
Other exercises in this chapter
Problem 4
Find the quotient and the remainder when the first integer is divided by the second. $$15,23$$
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Prove that the given predicate \(\mathrm{P}(n)\) in each algorithm is a loop invariant. Prove that the given predicate \(\mathrm{P}(n)\) in each algorithm is a
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Let \(A\) be a square matrix of order \(n .\) Let \(s_{n}\) denote the number of swappings of elements needed to find the transpose \(A^{\mathrm{T}}\) of \(A .\
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