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Chapter 7: Problem 7

Evaluate the arithmetic series. $$ \sum_{m=1}^{80}(4+5 m) $$

Short Answer

Expert verified
The sum of the first 80 terms of the arithmetic series \(4 + 5m\) is \(S_{80} = 16520\).

Step by step solution

01

Find the first term (a_1)

To find the first term, plug in \(m = 1\) into the formula: \(a_1 = (4+ 5(1)) = 4 + 5 = 9\)
02

Find the nth term (a_n)

To find the 80th term, plug in \(m = 80\) into the formula: \(a_n = (4 + 5(80)) = 4 + 400 = 404\)
03

Use the formula for the sum of an arithmetic series

Now that we have \(a_1, a_n\), and n, we can plug these into the formula for the sum of an arithmetic series: \(S_n = \frac{n}{2}(a_1 + a_n)\), \(S_{80} = \frac{80}{2}(9 + 404)\), \(S_{80} = 40(413)\), \(S_{80} = 16520\)
04

Conclusion

The sum of the arithmetic series is \(S_{80} = 16520\).

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