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]\begin{array}{l}
a_1=\square \
a_2=\square \
a_3=\square \
a_4=\square
\end{array}[/tex]
Answer (2)
Substitute k = 1 , 2 , 3 , 4 into the expression − 3 + 9 k .
Calculate a 1 = − 3 + 9 ( 1 ) = 6 .
Calculate a 2 = − 3 + 9 ( 2 ) = 15 .
Calculate a 3 = − 3 + 9 ( 3 ) = 24 .
Calculate a 4 = − 3 + 9 ( 4 ) = 33 .
The terms of the series are a 1 = 6 , a 2 = 15 , a 3 = 24 , a 4 = 33 .
Explanation
Understanding the Problem We are given the arithmetic series S 4 = ∑ k = 1 4 ( − 3 + 9 k ) . Our goal is to find the first four terms of this series, which are a 1 , a 2 , a 3 , and a 4 .
Finding the Terms To find the terms of the series, we will substitute k = 1 , 2 , 3 , 4 into the expression − 3 + 9 k .
Calculating a_1 For k = 1 , we have a 1 = − 3 + 9 ( 1 ) = − 3 + 9 = 6 .
Calculating a_2 For k = 2 , we have a 2 = − 3 + 9 ( 2 ) = − 3 + 18 = 15 .
Calculating a_3 For k = 3 , we have a 3 = − 3 + 9 ( 3 ) = − 3 + 27 = 24 .
Calculating a_4 For k = 4 , we have a 4 = − 3 + 9 ( 4 ) = − 3 + 36 = 33 .
Final Answer Therefore, the first four terms of the series are a 1 = 6 , a 2 = 15 , a 3 = 24 , and a 4 = 33 .
Examples
Arithmetic series are useful in many real-world applications. For example, consider a savings plan where you deposit a fixed amount of money each month. If you deposit $100 in the first month, $110 in the second month, $120 in the third month, and so on, the total amount you save over a certain period forms an arithmetic series. Understanding arithmetic series helps you calculate the total savings, plan your finances, and predict future growth. This concept is also applicable in calculating loan repayments, analyzing investment returns, and modeling various linear growth scenarios.
The first four terms of the arithmetic series are 6, 15, 24, and 33, derived from the expression − 3 + 9 k by substituting k = 1 , 2 , 3 , and 4 .
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