Problem 5 - Discrete Mathematics with Applications

Question

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

Answer

In conclusion, the total number of gifts sent on each day of Christmas up to the nth day can be calculated using the formula \(\sum_{i=1}^{n}(2 i-1)=n^{2}\). We have proved this formula by using the method of induction, demonstrating it to be correct for both the base case and the inductive step.

Step 1: Understand the given formula

The given formula represents the sum of the first n odd numbers, which can be written as \((2 i-1)\), where i is the index from 1 to n.

Step 2: Write the explicit formula for odd numbers

We can write the explicit formula for the sum of the first n odd numbers as follows: \[S_n = 1 + 3 + 5 + \dots + (2n - 1)\]

Step 3: Prove the given formula using induction

We need to prove \(\sum_{i=1}^{n}(2 i-1)=n^{2}\), using the method of induction. In order to do this, we need to prove two things: - Base case: The formula holds for the smallest value (n = 1) - Inductive step: If the formula holds for an arbitrary value n, then it also holds for n + 1

Step 4: Verify the base case

For n = 1, \[\sum_{i=1}^{1}(2 i-1) = (2(1)-1) = 1\] Since the sum of the first odd number is equal to the square of 1, the base case is true.

Step 5: Assume the inductive hypothesis for an arbitrary value n

Assume the given formula holds for an arbitrary value n, meaning that: \[\sum_{i=1}^{n}(2 i-1) = n^2\]

Step 6: Prove the formula for the next value n + 1

Now, we need to show that the formula holds for n + 1. We can apply the inductive step as follows: \[\sum_{i=1}^{n+1}(2 i-1) = \sum_{i=1}^n(2i-1) + (2(n+1) - 1)\] By the inductive hypothesis, \(\sum_{i=1}^{n}(2 i-1) = n^2\). So, we can substitute this into the above equation: \[n^2 + (2(n+1) - 1) = n^2 + 2n+ 1\] The equation of the right side can be factorized as: \[(n+1)^2\] Thus, we have proven the formula holds for n + 1. So using the method of induction, the given formula is correct.

Step 7: Conclusion

We have proved that the formula \(\sum_{i=1}^{n}(2 i-1)=n^{2}\) is correct for counting the total number of gifts sent on each day of Christmas up to the nth day.

Key Concepts

Sum of Odd Numbers

The concept of the sum of odd numbers is a useful tool in mathematics. By understanding it, you can solve various arithmetic problems. An odd number is any integer that is not divisible by 2, such as 1, 3, 5, and so on. In our problem, we're considering the sequence of the first 'n' odd numbers.
To express this in a formula, we use \(2i-1\). Here, 'i' represents a counting integer starting from 1 up to 'n'. This formula ensures each resultant number is an odd number.
The remarkable thing about the sum of the first 'n' odd numbers is that it always equals \(n^2\), a perfect square. This means if you were to add the first 3 odd numbers (1, 3, and 5), the result would be 9, which is \(3^2\).
Understanding this concept is fundamental when tackling a variety of mathematical challenges involving sequences and series.

Inductive Hypothesis

The inductive hypothesis is key in mathematical induction, a powerful proof technique. It involves two main steps.
First, we assume the given statement holds true for an arbitrary case 'n'. In our problem, this is expressed as \(\sum_{i=1}^{n}(2i-1) = n^2\). This assumption is what we call the inductive hypothesis.
This step is foundational for the next phase. We presume it's true to prove it also holds for the next integer, 'n + 1'.
🎈 **Key Steps of Inductive Hypothesis:**

• Assume the proposition is valid for 'n'.
• Express this assumption clearly.
• Use it as a basis for proving the statement for 'n + 1'.

Understanding and correctly applying the inductive hypothesis is crucial in proving statements universally.

Base Case

Starting with the base case is the crucial first step in a proof by induction. It serves as your proof foundation.
The base case involves showing that the statement you are trying to prove holds true for the initial value. In most problems, this value is 1. The base case is specific to your sequence or problem, such as proving \(\sum_{i=1}^{1}(2i-1) = 1^2\) as true.
Demonstrating the base case is essential because it establishes that the pattern or formula you're proving begins correctly. Without a solid base, the entire inductive proof cannot stand.
🧩 **Steps to Verify Base Case:**

• Identify the smallest 'n' in your sequence or problem.
• Prove the formula or statement holds true for this 'n'.

Once the base case is verified, you can confidently move to the inductive step, secure in that starting point.