Problem 17 - Discrete Mathematics with Applications

Question

Determine if each is true or false. $$\sum_{i=m}^{n} i=\sum_{i=m}^{n}(n+m-i)$$

Step-by-Step Solution

Answer

The statement is true, as for each term in the first summation, there is a corresponding term in the second summation such that their sum is equal to \(m + n\), hence the two summations are equal: \[ \sum_{i=m}^{n} i = \sum_{i=m}^{n}(n+m-i) \]

Step 1: Write the terms of the summations explicitly to visualize the sums

First, let's write out a few terms of each summation to get an idea of what they look like: The first sum: \[ \sum_{i=m}^{n} i = m + (m +1) + (m + 2) + \cdots + (n - 1) + n \] The second sum: \[ \sum_{i=m}^{n} (n+m -i) = (n+m - m) + (n+m - (m+1)) + (n+m - (m+2)) + \cdots + (n+m - n) \]

Step 2: Simplify the terms of the second summation

Now, let's simplify the terms in the second summation: \[ \sum_{i=m}^{n} (n+m -i) = n + (n-1) + (n-2) + \cdots + (m+1) + m \]

Step 3: Connect pairs of terms in the two summations

We can observe that there is a connection between each term in the first and second summations: \[ m + (m +1) + \cdots + n = n + (n - 1) + \cdots + m \] For each term "i" in the first summation, there is a corresponding term in the second summation for which their sum is equal to \(m + n\). This is true for every term from both summations.

Step 4: Conclude the equivalence of the summations

Since for each term in the first summation, there is a corresponding term in the second summation such that their sum is equal to \(m + n\), we can safely conclude that the two summations are indeed equal: \[ \sum_{i=m}^{n} i = \sum_{i=m}^{n}(n+m-i) \] So, the statement is true.

Key Concepts

Summation

Summation, in mathematics, is the operation of adding a sequence of numbers or expressions. It is denoted by the Greek letter sigma (\( \sum \)). Summation has a structure where you sum elements from a sequence, defined by a formula under the sigma notation.
For example, given an expression \( \sum_{i=1}^{n} a_i \), it implies you add all terms starting from \( a_1 \) up to \( a_n \). The lower and upper bounds, in this case \( 1 \) and \( n \), define the range of numbers for the index \( i \). We calculate and add each term's value as \( i \) changes from the starting index to the ending index.

• The process involves writing terms explicitly, which aids in identifying patterns or simplifying them for evaluation.
• It is a fundamental operation in calculus, linear algebra, and many mathematical proofs.

By understanding the nature and rules of summation, students can interpret and construct lawful mathematical statements.

Equivalence of Summations

The concept of equivalence of summations refers to two different summation expressions that yield identical results when evaluated. This point often arises in simplifying or transforming expressions, especially in scenarios involving symmetry or known identities.
In the given problem, to show the equivalence \[\sum_{i=m}^{n} i = \sum_{i=m}^{n}(n+m-i)\]we assess the structure of each summation. By simplifying:

• We observe that for each term in the first summation \( i \), there is a parallel term \( n+m-i \) in the second summation.
• These terms are paired such that their combined sum is always \( n+m \), maintaining balance across both expressions.

This systematic connection illustrates how changes in the sequence indices can be used to demonstrate equivalence. Such relationships are valuable in mathematical proofs and arguments, where strategic rearrangements or pairing of terms can reveal properties like equality.

Discrete Mathematics

Discrete Mathematics is a branch of math dealing with countable, distinct elements. It contrasts with continuous mathematics, which concerns uncountable or continuous elements. Topics in discrete math include logic, set theory, graph theory, algorithms, and number theory.
In this example of summation equivalence, discrete mathematics plays a crucial role. It provides the framework for discussing sequences and finite sums:

• Techniques used in discrete math include combinatorial arguments, bijective proofs, and induction, which often accompany summation problems to prove statements or explore properties.
• Discrete techniques help in assessing when two expressions are equal by counting or rearranging terms methodically, improving accuracy and understanding in mathematical operations.

Understanding topics within discrete mathematics equips students to solve complex problems involving finite structures and sequences, which are widespread in computer science, cryptography, and network theory.