Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-28: Find the sum. $$ \sum_{k=1}^{18}\left(\frac{1}{2} k+7\right) $$

Step-by-Step Solution

Answer

The sum is 211.5.

Step 1: Identify the Sum Formula

The problem asks for the sum of a sequence using the expression \( \sum_{k=1}^{18}\left(\frac{1}{2} k+7\right) \). This indicates we need to find the sum of the values generated by the expression \( \frac{1}{2} k + 7 \) as \( k \) ranges from 1 to 18.

Step 2: Simplify the Summation

Notice that the expression \( \frac{1}{2}k + 7 \) can be split based on summation rules: \( \sum_{k=1}^{18} \left(\frac{1}{2}k + 7\right) = \sum_{k=1}^{18} \frac{1}{2}k + \sum_{k=1}^{18} 7 \). This allows us to address each part separately.

Step 3: Calculate the First Summation

For \( \sum_{k=1}^{18} \frac{1}{2}k \), we can factor out the constant: \( \frac{1}{2} \sum_{k=1}^{18} k \).Using the formula for the sum of the first \( n \) natural numbers, \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \), we substitute \( n=18 \): \( \sum_{k=1}^{18} k = \frac{18 \times 19}{2} = 171 \).Thus, \( \sum_{k=1}^{18} \frac{1}{2}k = \frac{1}{2} \times 171 = 85.5 \).

Step 4: Calculate the Second Summation

For \( \sum_{k=1}^{18} 7 \), since 7 is constant for each \( k \), the summation simplifies to \( 18 \times 7 = 126 \).

Step 5: Combine the Results

Add the results from Step 3 and Step 4: \( 85.5 + 126 = 211.5 \). This is the sum of the series.

Key Concepts

Algebra

Algebra serves as the foundation for understanding mathematical expressions and equations. In this exercise, we utilize algebra to manipulate the summation expression \( \sum_{k=1}^{18} \left( \frac{1}{2} k + 7 \right) \). When approaching this problem, our goal is to simplify and break down the expression to make it easier to compute.
\( \frac{1}{2} k + 7 \) is a linear expression and its simplicity allows us to apply algebraic properties, like distributive and associative laws.
Through these algebraic rules, we split the summation of \( \frac{1}{2} k + 7 \) into two separate parts:

• \( \sum_{k=1}^{18} \frac{1}{2}k \)
• \( \sum_{k=1}^{18} 7 \)

Using such simplifications is a common algebraic technique to simplify problems and avoid cumbersome calculations. By learning how to apply these, we become adept at handling more complex algebraic equations too. This basic algebraic manipulation is a useful skill for any mathematician.

Summation Formula

The summation formula is pivotal for calculating the sum of arithmetic series quickly and efficiently. When dealing with the exercise, we used a particular summation formula that helps find the sum of natural numbers: \[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]In our problem, this formula was essential to calculate \( \sum_{k=1}^{18} k \).
The formula is derived from the observation that pairs of numbers from 1 to \( n \) always sum up to \( n+1 \). For 18, we used:

• 18 pairs of numbers sum to 19, resulting in a total of 171 when combined correctly.
• This allows a clean break where common factors, like \( \frac{1}{2} \), can be easily applied.

The other part of the expression, \( \sum_{k=1}^{18} 7 \), is simpler as the constant 7 repeats for each term. Multiplying 7 by 18 gives us an easy solution of 126.
The summation formula not only simplifies calculations but also gives deeper insights into patterns that exist within sequences.

Sequence and Series

Sequences and series are fundamental concepts in mathematics, helping to understand ordered sets of numbers and their summed values. In this exercise, the series represented by \( \sum_{k=1}^{18} \left( \frac{1}{2} k + 7 \right) \) is an arithmetic series.
This means each term increases by a consistent amount, making calculations predictable. The formula we've used helps in finding the sum of such well-structured series, which is more straightforward compared to more complex series.
Key characteristics of an arithmetic series:

• Each term after the first is generated by adding a constant to the preceding term.
• The expression between summation limits (1 to 18 in our case) defines the particular series we examine.

By understanding this sequence, we're able to decompose and solve problems step-by-step. Recognizing the type of sequence, such as arithmetic, is crucial for determining the right formulas to apply and for understanding the behavior of the series as a whole.