What is the value of the arithmetic series below?
[tex]S_{19}=\sum_{k=1}^{19}(4-3 k)[/tex]
Answer (2)
Expand the summation: S 19 = ∑ k = 1 19 4 − ∑ k = 1 19 3 k = 4 ∑ k = 1 19 1 − 3 ∑ k = 1 19 k .
Use the formula for the sum of the first n integers: ∑ k = 1 n k = 2 n ( n + 1 ) . In our case, n = 19 .
Substitute n = 19 into the formula: ∑ k = 1 19 k = 2 19 ( 19 + 1 ) = 2 19 ( 20 ) = 19 ( 10 ) = 190 .
Calculate the final result: S 19 = 4 ( 19 ) − 3 ( 190 ) = 76 − 570 = − 494 .
Explanation
Problem Analysis We are given the arithmetic series S 19 = ∑ k = 1 19 ( 4 − 3 k ) and we want to find its value.
Expanding the Summation We can expand the summation as follows: S 19 = k = 1 ∑ 19 ( 4 − 3 k ) = k = 1 ∑ 19 4 − k = 1 ∑ 19 3 k = 4 k = 1 ∑ 19 1 − 3 k = 1 ∑ 19 k
Using Summation Formulas We know that ∑ k = 1 n 1 = n and ∑ k = 1 n k = 2 n ( n + 1 ) . In our case, n = 19 . Therefore, k = 1 ∑ 19 1 = 19 and k = 1 ∑ 19 k = 2 19 ( 19 + 1 ) = 2 19 ( 20 ) = 19 ( 10 ) = 190
Calculating the Value of the Series Substituting these values back into the expression for S 19 , we get: S 19 = 4 ( 19 ) − 3 ( 190 ) = 76 − 570 = − 494
Final Answer Therefore, the value of the arithmetic series is − 494 .
Examples
Arithmetic series can be used to model situations where there is a constant increase or decrease. For example, if you save $4 on the first day and then save $3 less each subsequent day, the arithmetic series can help you calculate your total savings after 19 days. In this case, the sum would be negative, indicating a net loss rather than a gain. Understanding arithmetic series helps in financial planning, inventory management, and predicting linear trends.
The value of the arithmetic series S 19 = ∑ k = 1 19 ( 4 − 3 k ) is − 494 . This is calculated by expanding the summation, applying summation formulas, and performing the necessary calculations. Thus, the final result is S 19 = − 494.
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