Problem 10 - Discrete Mathematics with Applications

Question

Evaluate each sum. $$\sum_{k=1}^{5}(3-2 k) k$$

Step-by-Step Solution

Answer

The value of the summation \(\sum_{k=1}^{5}(3-2k)k\) is -65. We calculated this by evaluating the product of k and (3-2k) for each value of k from 1 to 5 and then summing up the results.

Step 1: Understand the range of summation

The summation runs for \(k = 1\) to \(k = 5\). We will calculate the sum of the products for these values.

Step 2: Calculate the products for each value of k

For each value of k in the range, we need to find the product of k and (3-2k). Let's calculate these products: For \(k = 1\): \(k(3-2k) = 1(3-2(1)) = 1(1) = 1\) For \(k = 2\): \(k(3-2k) = 2(3-2(2)) = 2(-1) = -2\) For \(k = 3\): \(k(3-2k) = 3(3-2(3)) = 3(-3) = -9\) For \(k = 4\): \(k(3-2k) = 4(3-2(4)) = 4(-5) = -20\) For \(k = 5\): \(k(3-2k) = 5(3-2(5)) = 5(-7) = -35\)

Step 3: Sum up the calculated products

Now that we have calculated the products for each value of k in the range, we simply need to sum them up: Sum = \(1 - 2 - 9 - 20 -35 = -65\)

Step 4: Write the final answer

The value of the given summation is -65. Therefore, \[ \sum_{k=1}^{5}(3-2k)k = -65 \]

Key Concepts

Series Evaluation

In discrete mathematics, series evaluation involves finding the sum of a sequence of terms. Let's break down this concept using the given exercise, where we need to evaluate \( \sum_{k=1}^{5}(3-2k)k \). A series is the sum of the values of a sequence defined by a formula for its terms.

To evaluate a series:

• Recognize the expression within the summation symbol \( \Sigma \), which indicates the formula for each term in the sequence.
• Identify the variable (here, \( k \)) and its range, which dictates the series of terms to sum.
• Compute each term as dictated by the formula and sum them up to reach the total.

In this exercise:

• The given range \(1 \leq k \leq 5\) tells us to compute the terms from \( k = 1 \) to \( k = 5 \).
• The expression \((3 - 2k)k \) determines the value of each term.

Evaluating the series, we first calculate each term and then sum them to identify the total, which ultimately gives us \(-65\) for this problem.

Mathematical Problem Solving

Mathematical problem solving in this context requires breaking down a summation into manageable steps. Here’s a straightforward strategy to tackle problems like the one given:

First, carefully interpret the sum to understand what needs to be calculated. This involves recognizing the role of each part of the summation notation. In this exercise, we have \((3-2k)k\) as our expression, with \(k\) ranging from 1 to 5.

Next, calculate each term individually within the specified range. For each integer value of \(k\), substitute \(k\) into \((3-2k)k\). This requires understanding basic arithmetic operations to find:

• \(k = 1\) gives 1
• \(k = 2\) gives -2
• \(k = 3\) gives -9
• \(k = 4\) gives -20
• \(k = 5\) gives -35

Finally, sum these individual results to arrive at the overall solution. This methodical approach ensures that errors are minimized and all aspects of the sum are correctly evaluated, which leads to the final solution: \(-65\). Effective problem-solving involves simple arithmetic and organization of calculations.

Discrete Mathematics Techniques

Discrete mathematics encompasses various techniques essential for analyzing mathematical structures. In the context of series evaluation, it emphasizes breaking down complex problems into smaller, manageable parts. This makes series evaluation in discrete mathematics systematic and organized.

Several techniques can be employed:

• Step-by-step computation - as we did by calculating each term individually for \(k=1\) to \(k=5\).
• Use of formulas to determine terms - understanding expressions like \((3-2k)k\) helps predict how varying \(k\) changes the result.
• Summation techniques - collecting calculated results into a coherent answer.

These techniques ensure accuracy and foster a deeper understanding of mathematical processes. They are crucial in fields that require precise calculations and logical reasoning. Practicing these discrete mathematics techniques by evaluating different forms of series deepens one's mathematical proficiency and problem-solving skills.