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Chapter 22: Problem 665

Find the sum of the arithmetic series \(5+9+13+\ldots+401\)

Short Answer

Expert verified
The sum of the arithmetic series \(5+9+13+\ldots+401\) is 20,300.

Step by step solution

01

- Find the number of terms in the series (n)

To find the number of terms in the series, we can use the formula n = ((last term - first term) / common difference) + 1. In this series, the first term (a) is 5, the last term (l) is 401, and the common difference (d) is 4. n = ((401 - 5) / 4) + 1 Calculating the number of terms we get: n = (396 / 4) + 1 n = 99 + 1 n = 100 So, there are 100 terms in the series.
02

- Apply the sum of the arithmetic series formula

The sum of an arithmetic series can be found with the formula: Sum = (n * (a + l))/2. In this case, we have: Sum = (100 * (5 + 401))/2
03

- Solve for the sum

Now, we can plug in our values and solve for the sum: Sum = (100 * (5 + 401))/2 Sum = (100 * 406)/2 Sum = 20300 The sum of the arithmetic series is 20,300.

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