Answer each question about the following arithmetic series: [tex]S_4=\sum_{k=1}^4(-3+9 k)[/tex]
What are the terms of the series?
[tex]a_1=[/tex]
[tex]a_2=[/tex] 15
[tex]a_3=[/tex] 24
[tex]a_4=[/tex] 33
What is the common difference of the arithmetic sequence on which the series is based?
-3
4
-6
9
What is the value of the arithmetic series?
[tex]S_4=[/tex]
Answer (2)
Calculate the first term: a 1 = − 3 + 9 ( 1 ) = 6 .
Verify the common difference: d = a 2 − a 1 = 15 − 6 = 9 .
Calculate the sum of the series: S 4 = 6 + 15 + 24 + 33 = 78 .
The value of the arithmetic series is 78 .
Explanation
Understanding the Problem We are given an arithmetic series S 4 = ∑ k = 1 4 ( − 3 + 9 k ) . We need to find the terms of the series, the common difference, and the value of the series.
Finding the First Term First, let's find the first term a 1 by substituting k = 1 into the expression − 3 + 9 k :
a 1 = − 3 + 9 ( 1 ) = − 3 + 9 = 6 So, a 1 = 6 .
Finding the Common Difference We are given a 2 = 15 , a 3 = 24 , and a 4 = 33 . The common difference d is the difference between consecutive terms. Let's check the difference between the given terms: a 2 − a 1 = 15 − 6 = 9 a 3 − a 2 = 24 − 15 = 9 a 4 − a 3 = 33 − 24 = 9 The common difference is indeed d = 9 .
Finding the Value of the Series Now, let's find the value of the arithmetic series S 4 . This is the sum of the first four terms: S 4 = a 1 + a 2 + a 3 + a 4 = 6 + 15 + 24 + 33 S 4 = 78 So, the value of the arithmetic series is S 4 = 78 .
Final Answer Therefore, the terms of the series are a 1 = 6 , a 2 = 15 , a 3 = 24 , a 4 = 33 . The common difference is 9 , and the value of the arithmetic series is 78 .
Examples
Arithmetic series are useful in many real-life situations. For example, consider a savings plan where you deposit a fixed amount each month. The total amount saved over time forms an arithmetic series. Understanding how to calculate the sum of an arithmetic series can help you predict your total savings after a certain period. Another example is calculating the total cost of installing fence posts where the cost increases for each subsequent post due to increasing distance or difficulty.
The terms of the series are a 1 = 6 , a 2 = 15 , a 3 = 24 , and a 4 = 33 . The common difference is 9 , and the sum of the series is S 4 = 78 .
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