Problem 61 - Beginning and Intermediate Algebra

Question

a) Evaluate \(\sum_{i=1}^{10}(2 i+7)\) by writing out each term and finding the sum. b) Evaluate \(\sum_{i=1}^{10}(2 i+7)\) using a formula for \(S_{n}\) c) Which method do you prefer and why?

Step-by-Step Solution

Answer

a) When listing each term and finding the sum manually, we get \(9 + 11 + 13 + \cdots + 27 = 180\). b) Using the arithmetic series formula, \(S_{10} = \frac{10(9 + 27)}{2} = 180\). c) The preferred method is the arithmetic series formula, as it is more efficient, especially with a large number of terms.

Step 1: a) Expand the series and calculate the sum manually.

To evaluate the sum of the series manually, we can begin by listing out each term. In this case, we are summing the first ten terms, where each term is given by the function \((2i + 7)\). So let's list out each term: Term 1: \(2(1) + 7 = 9\) Term 2: \(2(2) + 7 = 11\) Term 3: \(2(3) + 7 = 13\) . . . Term 10: \(2(10) + 7 = 27\) Now, we can sum up these terms: \[9 + 11 + 13 + \cdots + 27\] Because this is an arithmetic series, we can find the sum by using the formula \(S_n = \frac{n(a_1 + a_n)}{2}\). In this case, \(n = 10\), \(a_1 = 9\), and \(a_n = 27\). Plugging these values into the formula, we get: \(S_{10} = \frac{10(9 + 27)}{2}\) Simplifying, we have: \(S_{10} = \frac{10(36)}{2}\) \(S_{10} = \frac{360}{2}\) \(S_{10} = 180\) So, the sum of the first ten terms of the series is 180.

Step 2: b) Evaluate the sum using a formula for \(S_n\)

Next, let's evaluate the sum using a formula for the sum of an arithmetic series \(S_n\). To do this, we need to find the general formula for the series. As given, the formula for the series is \(2i+7\). To use the formula for \(S_n\), we need to recognize that this is an arithmetic series with a constant difference. The difference in this case is 2, as we can see from the formula. The first term, \(a_1\), is 9, as we found out in part a). Now, we can use the formula for an arithmetic series: \[S_n = \frac{n(a_1 + a_n)}{2}\] In this case, \(n = 10\), \(a_1 = 9\), and \(a_n = 27\). Plugging these values into the formula (like we did in part a)), we get: \(S_{10} = \frac{10(9 + 27)}{2}\) \(S_{10} = \frac{10(36)}{2}\) \(S_{10} = \frac{360}{2}\) \(S_{10} = 180\) So, the sum of the first ten terms of the series is 180 using the formula for \(S_n\).

Step 3: c) Preferred method and reason.

Comparing the two methods, it is clear that using the formula for the sum of an arithmetic series \(S_n\) is generally preferred. This is because it is faster and more efficient, especially when dealing with a large number of terms. Writing out each term manually becomes tedious and time-consuming as the number of terms increases. The formula provides a direct approach to finding the sum without the need to list out all the terms individually.

Key Concepts

Sum of Arithmetic Series

Understanding the sum of an arithmetic series is crucial for efficiently calculating lengthy sequences. An arithmetic series is a sequence where each term increases by a constant difference. For example, in the series given, each term is derived from the expression \(2i + 7\), resulting in a consistent increment.To find the sum, we can use the formula:

• \[S_n = \frac{n(a_1 + a_n)}{2}\]

Where:

• \(n\) is the number of terms
• \(a_1\) is the first term
• \(a_n\) is the last term

In this exercise, we find \(a_1 = 9\) and \(a_{10} = 27\), resulting in \(S_{10} = 180\).
This approach eliminates the need to calculate each term individually, providing a clean and efficient solution.

Arithmetic Sequence Formula

The arithmetic sequence formula helps identify the pattern of the sequence, which makes calculating sums much easier. When dealing with an arithmetic sequence, you find that each term gets its value from a specific expression. Here, the sequence is defined by:

• Expression: \(2i + 7\)
• Common Difference (d): 2

With such a formula, you can quickly determine any term without listing each one, as the constant difference helps connect terms. Quickly finding terms allows easy summation, particularly useful for larger sequences.
The formula for any term is important:

• \[a_i = a_1 + (i-1) \cdot d\]

This helps identify the structure of the sequence effectively.

Evaluating Series

When evaluating series, there are typically two approaches: manually listing terms or using formulas. In this case, evaluating \(\sum_{i=1}^{10}(2i+7)\) manually involves substituting each value of \(i\) from 1 to 10 to list terms before summing. This method is straightforward for small sequences but tedious for many terms.
Alternatively, using the sum formula for arithmetic series provides a streamlined approach:

• Find the sum without listing every term
• Faster and efficient, particularly for long series

Using the sum formula not only speeds up the calculation but also reduces the chance for errors in large sequences.

Mathematical Methods in Algebra

Mathematical methods in algebra often focus on simplifying calculations and reducing manual effort. In algebra, identifying patterns or formulas like those for arithmetic sequence sums can vastly improve efficiency and understanding. These techniques aim to:

• Optimize calculations
• Reduce errors
• Save time

Using algebraic methods, such as applying the sum formula, allows for direct solutions and deeper comprehension of the relationships within sequences. This exercise highlights the power of algebra in managing orderly data and provides insight into the cohesive structure of sequences and series.