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Chapter 7: Problem 6

Evaluate the arithmetic series. $$ 300+293+286+\cdots+55+48+41 $$

Short Answer

Expert verified
The arithmetic series has a common difference of -7 and 38 terms. Applying the sum of an arithmetic series formula, we get the sum as \(S_{38} = 6479\).

Step by step solution

01

1. Calculate the Common Difference

The common difference "d" can be found by subtracting the consecutive terms in the arithmetic series. In this case: $$ d = 293 - 300 = -7 $$
02

2. Determine the Number of Terms

To find the number of terms "n" in the arithmetic series, we can use the formula: $$ n = \frac{(a_n - a_1)}{d} + 1 $$ where \(a_1\) is the first term (300), \(a_n\) is the last term (41), and d is the common difference (-7). Using the values: $$ n = \frac{(41 - 300)}{-7} + 1 \approx 37.6 $$ Since the number of terms must be an integer, we conclude that there are 38 terms in the arithmetic series.
03

3. Apply the Sum of Arithmetic Series Formula

Now that we have the number of terms, we can use the formula for the sum of an arithmetic series: $$ S_n = \frac{n}{2}(a_1 + a_n) $$ Substitute the values: $$ S_{38} = \frac{38}{2}(300 + 41) $$
04

4. Calculate the Sum

Calculate the sum of the arithmetic series: $$ S_{38} = \frac{38}{2}(341) = 19 \cdot 341 = 6479 $$ Therefore, the sum of the given arithmetic series is 6479.

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