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Chapter 7: 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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