Problem 15 - Algebra 2

Question

Find the sum of each arithmetic series. $$ \sum_{k=3}^{7}(3 k+4) $$

Step-by-Step Solution

Answer

The sum of the series is 95.

Step 1: Identify the arithmetic series

The given series is \( \sum_{k=3}^{7} (3k + 4) \). We need to find the terms of this series starting from \( k=3 \) to \( k=7 \).

Step 2: Calculate each term of the series

Calculate the terms by substituting each value of \( k \) from 3 to 7 into the expression \( 3k + 4 \):- For \( k=3 \), the term is \( 3(3) + 4 = 9 + 4 = 13 \).- For \( k=4 \), the term is \( 3(4) + 4 = 12 + 4 = 16 \).- For \( k=5 \), the term is \( 3(5) + 4 = 15 + 4 = 19 \).- For \( k=6 \), the term is \( 3(6) + 4 = 18 + 4 = 22 \).- For \( k=7 \), the term is \( 3(7) + 4 = 21 + 4 = 25 \).

Step 3: Sum the terms

Add the terms calculated in Step 2:\( 13 + 16 + 19 + 22 + 25 = 95 \).

Key Concepts

Sum of Arithmetic Series

An arithmetic series is essentially the sum of numbers in an arithmetic sequence. In our specific example, the arithmetic series is given by the expression \( \sum_{k=3}^{7} (3k + 4) \). Let’s break this down in simpler terms. Here, each term follows a predictable pattern - they increase by the same amount. To find the sum of these terms:

• Identify the first and last term in the series. In our case, the first term is 13 and the last term is 25.
• Count the number of terms. Here, we have 5 terms (13, 16, 19, 22, 25).
• Use the formula for the sum of an arithmetic series: \( S_n = \frac{n}{2} \left( a + l \right) \), where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

Plugging the values into this formula gives us:\[ S_n = \frac{5}{2} \left( 13 + 25 \right) = \frac{5}{2} \times 38 = 95 \] Hence, the sum of this arithmetic series is 95.

Term Calculation

Calculating each term individually in the series \( \sum_{k=3}^{7} (3k + 4) \) is a systematic process. This involves substituting the integer values from 3 to 7 into the expression to get each term of the series.

• For \( k = 3 \): Substitute into the expression: \(3(3) + 4 = 13\)
• For \( k = 4 \): Calculate as \(3(4) + 4 = 16\)
• Continue similarly for \( k = 5, 6, 7 \) to get 19, 22, and 25 respectively.

Understanding this portion ensures you can break down any similar problem where the series is defined by a general formula, by substituting sequential values into the equation. The key takeaway is that each term is derived by following the pattern defined by the given expression, ensuring a list of numbers that form a simple sequence.

Series Limits

When working with a series like \( \sum_{k=3}^{7} (3k + 4) \), it's important to understand the concept of series limits. These are defined by the starting and ending points of the series, which are crucial to determine what terms are included in the sum.

• Lower limit: The index where the series begins, here it is \( k = 3 \).
• Upper limit: The index where the series ends, in this case, \( k = 7 \).

This series notation is a concise way to express a sequence of operations over a range of numbers. You perform a specific task (such as calculating each term) for each integer value between the lower and upper limits, inclusive. Comprehending series limits helps you to ensure you capture all numbers in the series, and allows you to accurately compute the total sum by including all designated terms.