Problem 62 - College Algebra with Modeling and Visualization

Question

Write out the terms of the series and then evaluate it. $$\sum_{k=1}^{6}(3 k-1)$$

Step-by-Step Solution

Answer

The sum of the series is 57.

Step 1: Identify the General Term

The given series is \( \sum_{k=1}^{6} (3k - 1) \). The general term here is \( 3k - 1 \). This expression gives us the individual terms of the sequence that we will sum.

Step 2: Substitute Values for k

Substitute values from \( k = 1 \) to \( k = 6 \) into the expression \( 3k - 1 \) to find each term:- For \( k = 1 \), term = \( 3(1) - 1 = 2 \)- For \( k = 2 \), term = \( 3(2) - 1 = 5 \)- For \( k = 3 \), term = \( 3(3) - 1 = 8 \)- For \( k = 4 \), term = \( 3(4) - 1 = 11 \)- For \( k = 5 \), term = \( 3(5) - 1 = 14 \)- For \( k = 6 \), term = \( 3(6) - 1 = 17 \)

Step 3: Write Out the Series

Now, write out the series using the calculated terms:\[ 2 + 5 + 8 + 11 + 14 + 17 \]

Step 4: Evaluate the Series

Add up all the terms in the series:\[ 2 + 5 + 8 + 11 + 14 + 17 = 57 \]

Step 5: State the Final Answer

After evaluating the sum of the series, the final answer to \( \sum_{k=1}^{6} (3k - 1) \) is \( 57 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are foundational in mathematics. They consist of variables, constants, and arithmetic operators like addition, subtraction, multiplication, and division. An algebraic expression can be as simple as a single variable or as complex as a combination of multiple terms and operations. In our exercise, the expression is \(3k - 1\).

• Variables: Symbols that represent unknown values. Here, \(k\) is a variable that changes its value within a given range.
• Constants: Fixed values that do not change. In \(3k - 1\), both 3 and 1 are constants.
• Operators: Symbols that denote mathematical operations such as \(-\), \(+\), \(\times\), and \(\div\). The '-' symbol in \(3k - 1\) represents subtraction.

Understanding how to construct and deconstruct algebraic expressions is critical in applying them within series, equations, and other mathematical concepts.

Arithmetic Series

An arithmetic series is a sum of terms in an arithmetic sequence where each term increases by a constant value, called the common difference. In the provided exercise, the sequence generated by substituting into \(3k - 1\) forms an arithmetic series: 2, 5, 8, 11, 14, 17.

• Common Difference: It is the difference between successive terms. For our series, the common difference is 3 \((5 - 2 = 3)\).
• First Term: This is the initial term of the series. Here it is 2.
• Number of Terms: Indicates how many terms are present in the series. In this case, we have six terms to sum.

To find the sum of an arithmetic series efficiently, one can use the formula: \( S_n = \frac{n}{2}(a + l) \), where \(S_n\) is the sum, \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term. Substituting these into the formula certainly speeds up calculations in larger sequences where manual addition is less practical.

Summation Notation

Summation notation, expressed with the Greek letter \(\Sigma\), provides a concise way to denote the sum of a sequence of terms. It specifies a formula for the general term and the index range over which the terms are summed. In the exercise, \(\sum_{k=1}^{6}(3k-1)\) denotes the sum of terms from \(k = 1\) to \(k = 6\), where each term follows the rule \(3k - 1\).

• Lower Limit: The starting value of the index, \(k = 1\) in this series.
• Upper Limit: The ending value of the index, \(k = 6\) here. It tells us the range of values \(k\) can take.
• General Term: The expression \(3k - 1\) in the notation defines each term generated from the values of \(k\).

This notation is not only a more compact way to illustrate the sum of terms but also helps in easily spotting patterns and efficiently performing calculations, which are especially valuable when dealing with long sequences.