Problem 9 - Discrete Mathematics with Applications

Question

Find the number of terms in the expansion of each expression. $$\left(\sum_{i=-2}^{5} a_{i}\right)\left(\sum_{i=-1}^{3} b_{i}\right)\left(\sum_{i=0}^{4} c_{i}\right)$$

Step-by-Step Solution

Answer

In the given expression, there are 8 terms in the first summation, 5 terms in the second summation, and 5 terms in the third summation. To find the total number of terms in the expansion, multiply the number of terms in each summation: \(8 \times 5 \times 5 = 200\). Therefore, there are 200 terms in the expansion of the expression.

Step 1: Analyze each summation individually

Firstly, we shall determine the number of terms in each summation before expanding the expression. For the first summation, we have $$ \sum_{i=-2}^{5} a_{i} $$ The index 'i' starts from -2 and goes up to 5. To find the total number of terms in this summation, we can simply count the numbers from -2 to 5 inclusively, or calculate it as: Number of terms in first summation = (Upper limit - Lower limit + 1) = (5 - (-2) + 1) = 8 For the second summation, we have $$ \sum_{i=-1}^{3} b_{i} $$ The index 'i' starts from -1 and goes up to 3. Similarly, we can find the total number of terms in this summation as follows: Number of terms in second summation = (3 - (-1) + 1) = 5 For the third summation, we have $$ \sum_{i=0}^{4} c_{i} $$ The index 'i' starts from 0 and goes up to 4. We can calculate the total number of terms in this summation as: Number of terms in third summation = (4 - 0 + 1) = 5

Step 2: Calculate the total number of terms in the expansion

Now that we know the number of terms in each summation, we can find the total number of terms in the expansion by multiplying the numbers of terms from each summation. Number of terms in the expansion = (Number of terms in first summation) × (Number of terms in second summation) × (Number of terms in third summation) Number of terms in the expansion = 8 × 5 × 5 Number of terms in the expansion = 200 So, there are 200 terms in the expansion of the given expression.

Key Concepts

Discrete Mathematics Problems

Discrete mathematics involves a set of mathematical topics that deals with objects that can assume only distinct, separated values. It is in contrast to continuous mathematics, which deals with objects that can vary smoothly. Discrete mathematics problems often include topics such as logic, set theory, combinatorics, graph theory, and in this case, sequences and series.

When solving problems like the given exercise, it is important to understand the fundamental definitions and operations involved in counting discrete elements. For instance, with summation sequences in discrete mathematics, we determine the total number of terms by considering the start and end values of the index, counting each possible value the index can assume. This type of problem is an essential skill for students to learn as they often form the basis for more complex problems in discrete mathematics.

Sequence and Series

In a sequence, we have an ordered list of numbers where each number is called a term. A series is essentially the summation of the terms of a sequence.

The problem presented requires an understanding of how to handle series that involve summation notation. This notation is compact and allows us to express the addition of terms that follow a pattern. The boundaries of the summation (lower limit and upper limit) dictate where the series begins and ends, while the term under the summation symbol defines the pattern or rule that generates the series' terms.

Calculating the number of terms in such a series involves finding the inclusive count from the lower to the upper limit, a concept cemented through the formula: Number of terms = (Upper limit - Lower limit + 1). Understanding this concept is crucial when dealing with discrete mathematics problems involving sequences and series.

Mathematical Induction

Mathematical induction is a powerful and fundamental proof technique used in discrete mathematics and beyond. While not directly applied in the given problem, understanding mathematical induction is key for tackling a wide range of discrete mathematics problems, especially those involving proofs.

Induction allows us to prove that a property or statement holds true for all natural numbers. It consists of two critical steps: the base case and the inductive step. The base case verifies that the statement holds for the initial value (often 0 or 1), while the inductive step proves that if the statement holds for some integer 'k', then it must also hold for 'k+1'. Through this process, we can infer that the statement is true for all natural numbers. This form of reasoning is akin to a domino effect; once the first domino falls (our base case), it ensures that all subsequent dominos fall too (the inductive step).