Problem 11 - Discrete Mathematics with Applications

Question

Evaluate each sum. $$\sum_{j=-1}^{4}(j-2)^{2}$$

Step-by-Step Solution

Answer

The sum of the squared values of \((j-2)^{2}\) for all integers j from -1 to 4 is 19: \(\sum_{j=-1}^{4}(j-2)^{2} = 9 + 4 + 1 + 0 + 1 + 4 = 19\).

Step 1: Evaluate each term for j in the range (-1, 4)

For each value of j in the range of (-1, 4), substitute it into the expression \((j-2)^{2}\) and compute the squared value. We have the following terms: \(j=-1: (-1-2)^{2}=(-3)^{2}=9\) \(j=0: (0-2)^{2}=(-2)^{2}=4\) \(j=1: (1-2)^{2}=(-1)^{2}=1\) \(j=2: (2-2)^{2}=(0)^{2}=0\) \(j=3: (3-2)^{2}=(1)^{2}=1\) \(j=4: (4-2)^{2}=(2)^{2}=4\)

Step 2: Calculate the sum of the squared values

Now, we need to add all the squared values together. The sum is given by the following expression: \(\sum_{j=-1}^{4}(j-2)^{2} = 9 + 4 + 1 + 0 + 1 + 4\)

Step 3: Evaluate the sum

Finally, add up all the squared values to get the answer: \(\sum_{j=-1}^{4}(j-2)^{2} = 9 + 4 + 1 + 0 + 1 + 4 = 19\) The sum of the squared values of (j-2) for all integers j from -1 to 4 is 19.

Key Concepts

Sigma Notation

Sigma notation, denoted by the Greek letter sigma \( \Sigma \), serves as a convenient and compact way of representing long sums. It is extensively used in discrete mathematics and calculus for summing a sequence of terms.

The structure of sigma notation includes an expression to be summed, an index of summation (typically \( i \), \( j \) or \( k \)), the lower limit, and the upper limit. For example, \( \sum_{j=-1}^{4}(j-2)^{2} \) tells us to sum the squares of \( j-2 \) starting when \( j \) equals -1 and ending when \( j \) equals 4.

To systematically use sigma notation:

• Identify the variable and its range; in our case, \( j \) ranges from -1 to 4.
• Understand the function to apply to each term; here, we square the result of \( j-2 \).
• Sum the function's results over the range of the variable.

Understanding sigma notation is key to working with series and sequences effectively, as it saves time and space, avoiding the need to write out all terms of the series explicitly.

Sequence Evaluation

Evaluating sequences involves finding the value of each term in a sequence given an explicit formula. For example, given the sequence defined by the formula \( (j-2)^{2} \) and the values of \( j \) ranging from -1 to 4, we calculate each term individually.

The step-by-step approach to this evaluation process means we substitute the values of \( j \) into the formula and compute the outcome:

• For \( j=-1 \: (-1-2)^{2} = 9 \)
• For \( j=0 \: (0-2)^{2} = 4 \)
• And so on for each value of \( j \) within the specified range.

Sequence evaluation is fundamental in understanding the behavior of sequences and is often a precursor to more advanced topics such as series sums and convergence. By mastering sequence evaluation, one gains a better understanding of how various sequences progress and change over their range.

Series Sum

In discrete mathematics, the sum of a series involves adding together a list of numbers derived from a sequence, usually represented in a formulaic form. The series sum for the given exercise equals 19, which is the result of summing the evaluated terms from the sequence.

To sum up a series correctly, follow these steps:

• Evaluate each term in the sequence using the formula provided, within the given range.
• Add all these evaluated terms to compute the series sum, ensuring that no terms are missed or duplicated.

In our case, the series sum of \( \sum_{j=-1}^{4}(j-2)^{2} = 9 + 4 + 1 + 0 + 1 + 4 \). This demonstrates the simplification power of sigma notation for expressing and evaluating the sum of a progression of terms, which otherwise can be cumbersome to write out. Series sums are central to many applications across mathematics and sciences because they allow for the aggregation of numerous quantities to form a single value.