Problem 52 - Contemporary Precalculus

Question

$$\text { Find the sum.}$$ $$\sum_{n=1}^{35}\left(\frac{2 n+4}{8}\right)$$

Step-by-Step Solution

Answer

Answer: The sum of the given arithmetic series is 175.

Step 1: Identify the first term, last term, and the number of terms

We know the general term is given by $$\frac{2n+4}{8}$$. We are asked to find the sum of the series from n = 1 to n = 35. Therefore, we have: - The first term, \(a_1 = \frac{2(1)+4}{8} = \frac{6}{8}\) - The last term, \(a_{35} = \frac{2(35)+4}{8}=\frac{74}{8}\) - The number of terms, n = 35

Step 2: Calculate the sum of the series

Now that we have the first term, last term, and the number of terms, we can use the sum formula: $$\text{Sum} = n\left(\frac{a_1 + a_n}{2}\right)$$ Plug in the values we found in step 1: $$\text{Sum} = 35\left(\frac{\frac{6}{8}+\frac{74}{8}}{2}\right)$$

Step 3: Simplify and find the sum

Simplify the expression inside the parentheses first by adding the fractions: $$\frac{6}{8}+\frac{74}{8} = \frac{80}{8} = 10$$ Now, divide the resulting sum by 2: $$\frac{10}{2}=5$$ Finally, multiply the number of terms (35) by the result we found (5): $$\text{Sum}=35 \times 5 = 175$$ So, the sum of the given series is 175.

Key Concepts

Sum of Series

The sum of a series is the total amount you get when you add up all the numbers in a sequence. In the context of arithmetic sequences, these are specific types of series where each term is generated by adding a constant value to the previous term. This makes it easy to calculate their sum using a formula.
For this exercise, the series consists of terms derived from the formula \(\frac{2n+4}{8}\). We're tasked to find the sum from \(n = 1\) to \(n = 35\).
The actual sum is calculated by finding the first term, the last term, and knowing the number of terms, then applying a formula that simplifies the process of adding all terms individually. This method is efficient and saves time, especially when working with sequences containing many numbers.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. This difference is called the common difference. In our exercise, the terms are generated from the expression \(\frac{2n+4}{8}\), where \(n\) represents the position of each term in the series.
To understand how this works, look at the numbers:

• First term: Set \(n = 1\) and calculate \(\frac{2(1) + 4}{8} = \frac{6}{8}\)
• Second term: Set \(n = 2\) to get \(\frac{2(2) + 4}{8} = \frac{8}{8}\)
• And so on until the 35th term.

You'll notice each step increases \(n\) by 1, influencing the numerator which alters the term's value. The regular progression of terms forms the arithmetic sequence, essential for determining the sum efficiently.

Series Formula

To find the sum of an arithmetic sequence, you don't have to add each number manually. Instead, use the formula:\[\text{Sum} = n\left(\frac{a_1 + a_n}{2}\right)\]where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.
For the exercise at hand, identify:

• First term \(a_1 = \frac{6}{8}\)
• Last term \(a_{35} = \frac{74}{8}\)
• Total number of terms \(n = 35\)

Substitute these into the formula:\[\text{Sum} = 35\left(\frac{\frac{6}{8} + \frac{74}{8}}{2}\right) = 35 \times 5\]
Thus, calculating the sum becomes straightforward. This formula is a powerful tool that leverages the properties of arithmetic sequences to give quick results.