Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{k=3}^{5}(5-4 k) $$

Step-by-Step Solution

Answer

The sum of the series is -33.

Step 1: List the Terms of the Series

First, we need to identify the terms of the series by substituting the values of \(k\) into the expression \(5 - 4k\). Write out each term separately for \(k = 3\), \(k = 4\), and \(k = 5\).

Step 2: Compute Each Term Separately

- For \(k = 3\), the term is \(5 - 4(3) = 5 - 12 = -7\).- For \(k = 4\), the term is \(5 - 4(4) = 5 - 16 = -11\).- For \(k = 5\), the term is \(5 - 4(5) = 5 - 20 = -15\).

Step 3: Write the Arithmetic Series

The arithmetic series as the sum of terms is: \(-7 + (-11) + (-15)\).

Step 4: Calculate the Sum of the Series

Now add up these terms: \(-7 + (-11) + (-15)\). First, calculate \(-7 - 11 = -18\).Next, calculate \(-18 - 15 = -33\).

Step 5: Conclusion

The sum of the arithmetic series is \(-33\).

Key Concepts

Sum of Series

The sum of a series involves adding up all the terms in a sequence. In the context of an arithmetic series, where each term is equally spaced from the previous one, calculating the sum is straightforward once you know the terms. Let’s break this down:

• Identify the individual terms in the series by substituting each corresponding index value, in our case, the variable \(k\).
• Write out these terms to visualize them as a list.
• Finally, add all the terms together to find the sum of the series.

In the given solution, the terms of the series \(-7, -11,\) and \(-15\) were found by substituting the respective values of \(k\).
Adding these together:

• Start with the sum of the first two terms: \(-7 + (-11) = -18\).
• Move to the subsequent addition: \(-18 + (-15) = -33\).

Thus, the sum of the given arithmetic series is \(-33\). Understanding how to calculate the sum pulls together the whole goal of using a series in arithmetic sequences.

Substitution

Substitution is a crucial method used in finding the terms of a series. This mathematical technique allows for replacing a variable with a specific number, making it vital for identifying exact values in sequences.Here’s how substitution works in our exercise:

• First, identify the expression representing each term. For this exercise, that expression is \(5 - 4k\).
• Choose the specific values for the variable \(k\), which are often given as a range. Here, \(k = 3, 4,\) and \(5\).
• For each value of \(k\), substitute it into the expression to find each term individually.

For instance:

• With \(k = 3\), substitution yields the term: \(5 - 4(3) = -7\).
• Through further substitution, you compute: \(5 - 4(4) = -11\) and \(5 - 4(5) = -15\).

Substitution aids in simplifying the series into concrete numbers, paving the way to calculate the sum effectively.

Arithmetic Sequence

An arithmetic sequence is a series of numbers with a distinct pattern: each term is created by adding or subtracting a constant to the previous term. Understanding this pattern is fundamental to dealing with arithmetic sequences and series effectively.In the exercise at hand:

• The expression \(5 - 4k\) generates the sequence when different values of \(k\) are substituted.
• This expression defines a linear function, reflecting the constant difference between terms. Here, each term reduces by \(4\) as \(k\) increases by \(1\).

Recognizing the characteristics of an arithmetic sequence helps:

• Identify the starting term of the series.
• Understand the step, or common difference, between terms, which is \(-4\) in this case.
• Predict other terms quickly by continuing the pattern.

The regular pattern of an arithmetic sequence not only simplifies finding the terms but also assists in calculating the total sum of the series — essential for solving many mathematical problems efficiently.