Problem 46 - Discrete Mathematics with Applications

Question

Evaluate each sum and product, where \(p\) is a prime and \(I=\\{1,2,3,5\\}.\) $$\sum_{j=1}^{4}\left(3^{j}-3^{j-1}\right)$$

Step-by-Step Solution

Answer

The sum of the given expression \(\sum_{j=1}^{4}\left(3^{j}-3^{j-1}\right)\) is equal to \(80\).

Step 1: Understand the given expression

We are given the following sum to evaluate: $$ \sum_{j=1}^{4}\left(3^{j}-3^{j-1}\right) $$ We need to calculate the expression \(3^j - 3^{j-1}\) for j = 1, 2, 3, and 4, and then add those terms up to get the final answer.

Step 2: Calculate the terms for each j value

Now, let's calculate the terms for each j value: 1. For j = 1: $$ 3^{1} - 3^{1-1} = 3 - 1 = 2 $$ 2. For j = 2: $$ 3^{2} - 3^{2-1} = 9 - 3 = 6 $$ 3. For j = 3: $$ 3^{3} - 3^{3-1} = 27 - 9 = 18 $$ 4. For j = 4: $$ 3^{4} - 3^{4-1} = 81 - 27 = 54 $$

Step 3: Add up the terms for each j value

Now that we have calculated the terms for each j value, let's add them up: $$ 2 + 6 + 18 + 54 = 80 $$ Thus, the sum of the given expression is 80.

Key Concepts

Evaluating Sums

Evaluating sums is a fundamental skill in mathematics, especially in discrete mathematics when dealing with sequences or series. A sum is simply the total of the values in a sequence added together. In mathematical notation, this is represented using the Greek sigma symbol For example, the sum given in the exercise is represented as \[\sum_{j=1}^{4}(3^{j}-3^{j-1})\]This means we need to calculate \((3^1 - 3^0) + (3^2 - 3^1) + (3^3 - 3^2) + (3^4 - 3^3)\) and then find their total. Breaking sums down into simpler steps allows you to handle complex expressions more easily. You evaluate each term one by one and then combine them to get the final result.
Understanding this process is crucial as it's applied in various mathematical and real-world contexts. Evaluating sums helps in predicting patterns, calculating costs, and even in computer science for algorithms.

Prime Numbers

Prime numbers play a crucial role in discrete mathematics and are essential for various applications such as cryptography. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
For example, the smallest prime number is 2, followed by 3, 5, 7, and so on. These numbers cannot be divided evenly by any other numbers except for 1 and the number itself.

• They are the building blocks of the integers as every number can be expressed as a product of primes.
• Prime numbers are infinite. There is no largest prime number.
• Understanding primes helps in writing efficient algorithms in computer science.

In the provided exercise, prime numbers were not directly evaluated, but knowledge of primes can help when determining sequences or identifying properties of mathematical expressions. Recognizing primes can simplify many mathematical problems and support deeper mathematical theories.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like plus or minus) that represent a particular value or set of values. They are fundamental in helping us to interpret and manipulate mathematical relationships clearly and logically.
In the provided exercise's expression \[3^j - 3^{j-1}\]mathematical operations are used to simplify or solve the problem. This expression involves exponential notation, where 3 is raised to the power of the variable \(j\), demonstrating the principle of mathematical execution order.

• Expressions can be simplified using arithmetic operations.
• Variables can create general cases for solving problems over a range of values.
• Understanding the different forms of expressions aids in solving complex problems by breaking them into manageable pieces.

Mathematical expressions allow us to model real-world situations and provide a concise way to express complex ideas. Mastery makes it easier to tackle higher mathematical concepts and operations effortlessly.