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Chapter 6: Problem 9

Evaluate the arithmetic series. $$ \sum_{k=5}^{65}(4 k-1) $$

Short Answer

Expert verified
The sum of the arithmetic series given by \(\sum_{k=5}^{65}(4 k-1)\) can be found using the arithmetic series formula. We have the first term \(a_1 = 19\), last term \(a_n = 259\), common difference \(d = 4\) and number of terms \(n = 61\). Using the formula, sum = \(\frac{(First\ term + Last\ term) \cdot Number\ of\ terms}{2} = \frac{(19 + 259) \cdot 61}{2} = 8469\). Thus, the sum of the given series is 8469.

Step by step solution

01

Find the number of terms in the series.

To find the number of terms of this arithmetic series, we can use the formula: Number of terms = (Last term - First term) / Common difference + 1 Since the series is given by \(\sum_{k=5}^{65}(4k - 1)\), our first term's index is 5, and our last term's index is 65. To find out the values for these terms, we will substitute k with 5 and 65: First term = \(4(5)-1 = 19\) Last term = \(4(65)-1 = 259\) Now we want to find the common difference. Looking at the equation \(4k - 1\), we see that if we increase k by 1, the term increases by 4 since the coefficient of k is 4. Therefore, the common difference is 4. Number of terms = \((259 - 19) / 4 + 1 = 61\)
02

Find the sum of the series using the arithmetic series formula.

We can use the following formula to find the sum of an arithmetic series: Sum = \(\frac{(First\ term + Last\ term) \cdot Number\ of\ terms}{2}\) Now, we can plug our values into the formula: Sum = \(\frac{(19 + 259) \cdot 61}{2} = \frac{278 \cdot 61}{2} = 8469\)
03

Write the final answer.

The sum of the arithmetic series given by \(\sum_{k=5}^{65}(4 k-1)\) is 8469.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Difference
In an arithmetic series, each term is obtained by adding a fixed number to the previous term. This fixed number is known as the "common difference". Understanding the common difference is crucial, as it allows you to recognize patterns and predict the behavior of an arithmetic sequence.

Let's break it down:
  • Identify the general formula for the series, which in this case is given by \(4k - 1\).
  • The coefficient of \(k\) in the expression \(4k - 1\) is 4.
  • This coefficient represents the common difference. Hence, our common difference for this series is 4.
With this information, you can easily determine subsequent terms of the series by repeatedly adding the common difference, starting from the first term. This systematic approach simplifies the process of analyzing and evaluating arithmetic sequences.
Sum of Arithmetic Series
Calculating the sum of an arithmetic series can seem daunting, but a clear formula simplifies the process. The formula for the sum \(S\) of an arithmetic series is:\[S = \frac{(\text{First term} + \text{Last term}) \times \text{Number of terms}}{2}\]

Let's guide you through this formula with our example:
  • The first term, when \(k = 5\), is \(19\).
  • The last term, when \(k = 65\), is \(259\).
  • The number of terms, as derived, is 61.
By plugging these values into the formula, we compute the sum:\[S = \frac{(19 + 259) \times 61}{2} = 8469\]This arithmetic sum efficiently captures the cumulative, orderly progression of the series.
Number of Terms in Series
To unravel the series effectively, it's important to determine how many terms it contains. The number of terms provides clarity on the sequence's depth.

Here's how to calculate it for an arithmetic series:
  • Use the formula: \((\text{Last term} - \text{First term}) / \text{Common difference} + 1\).
  • For our series, the first term is \(19\) and the last term is \(259\), derived from the expressions \(4 \cdot 5 - 1\) and \(4 \cdot 65 - 1\) respectively.
  • The common difference is 4.By solving, we find:\[(259 - 19) / 4 + 1 = 61\]
This formula provides an exact count, ensuring you have an accurate framework for summing or further analyzing the series.

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