Problem 3 - Discrete Mathematics with Applications

Question

Evaluate each sum. $$\sum_{j=0}^{4}(j-1)$$

Step-by-Step Solution

Answer

The short answer to evaluating the sum \(\sum_{j=0}^{4}(j-1)\) is as follows: Substitute the values of 'j' in the summand and simplify, then sum the simplified values. \((-1) + 0 + 1 + 2 + 3 = 5\). So, the evaluated sum is \(5\).

Step 1: Understand the given sum notation

We are given a sum notation: \[ \sum_{j=0}^{4}(j-1) \] Here, 'j' is the variable, and the range of the sum is from j=0 to j=4.

Step 2: Substitute the values of 'j' in the summand

Substitute 'j' with values in the range of 0 to 4: \[ (0-1) + (1-1) + (2-1) + (3-1) + (4-1) \]

Step 3: Simplify the summands

Simplify the summands by performing the subtraction operations inside the parentheses: \[ -1 + 0 + 1 + 2 + 3 \]

Step 4: Sum the simplified summands

Add the simplified summands together to get the final result: \[ -1 + 0 + 1 + 2 + 3 = 5 \] Therefore, the evaluated sum is \(5\).

Key Concepts

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying 'smoothly', the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly, but have distinct, separated values. Summation notation, a key topic in discrete mathematics, is a concise way of representing the addition of a sequence of values.

For instance, in the given exercise where we have to evaluate \( \sum_{j=0}^{4}(j-1) \), we're dealing with a finite series of integers and performing operations based on discrete steps. Each value of 'j' represents a discrete point, and we sum each corresponding \( j-1 \) to find the total. This fittingly illustrates how discrete mathematics structures, even something like summation notation, are founded on separated, countable values.

Series and Sequences

In mathematics, a sequence is an ordered list of numbers following some rule, while a series is the sum of a sequence of numbers. Understanding series and sequences is crucial for solving problems involving summation notation.

The exercise provided, \( \sum_{j=0}^{4}(j-1) \), can be considered a series. The rule here is each term in the sequence is given by \( j-1 \) for integer values of \( j \) from 0 to 4. We can write out the sequence of terms corresponding to this rule as -1, 0, 1, 2, 3. The series is then the sum of these terms. In solving the exercise, we demonstrate how we translate the general rule into a specific sequence and then into a series by summing the terms.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite number of cases, often involving sequences or series. It is based on proving a base case and then proving that if any one case is true, the next case is true as well. While the given exercise \( \sum_{j=0}^{4}(j-1) \) does not require the use of mathematical induction due to its finite nature, it's an important tool for understanding related concepts in discrete mathematics.

For example, if we had to prove a rule for an infinite series or the validity of a summation expression for all natural numbers, we would turn to mathematical induction. Typically, the process involves two steps: the base case test (\