Use the arithmetic series [tex]S _5=\sum_{k=1}^5(1+3 k)[/tex] to answer each question. What are the terms of this arithmetic series?
[tex]\begin{array}{l}
a_1=\square \
a_2=\square \
a_3=\square \
a_4=\square \
a_5=\square
\end{array}[/tex]
Answer (2)
Calculate the first term by substituting k = 1 into the expression 1 + 3 k , which gives a 1 = 4 .
Calculate the second term by substituting k = 2 into the expression 1 + 3 k , which gives a 2 = 7 .
Calculate the third term by substituting k = 3 into the expression 1 + 3 k , which gives a 3 = 10 .
Calculate the fourth term by substituting k = 4 into the expression 1 + 3 k , which gives a 4 = 13 .
Calculate the fifth term by substituting k = 5 into the expression 1 + 3 k , which gives a 5 = 16 .
The terms of the arithmetic series are: a 1 = 4 , a 2 = 7 , a 3 = 10 , a 4 = 13 , a 5 = 16 .
Explanation
Understanding the Arithmetic Series We are given the arithmetic series S 5 = ∑ k = 1 5 ( 1 + 3 k ) . Our goal is to find the first five terms of this series, which are a 1 , a 2 , a 3 , a 4 , and a 5 . Each term can be found by substituting the corresponding value of k into the expression 1 + 3 k .
Calculating the First Term To find the first term, a 1 , we substitute k = 1 into the expression 1 + 3 k :
a 1 = 1 + 3 ( 1 ) = 1 + 3 = 4 So, the first term is 4.
Calculating the Second Term To find the second term, a 2 , we substitute k = 2 into the expression 1 + 3 k :
a 2 = 1 + 3 ( 2 ) = 1 + 6 = 7 So, the second term is 7.
Calculating the Third Term To find the third term, a 3 , we substitute k = 3 into the expression 1 + 3 k :
a 3 = 1 + 3 ( 3 ) = 1 + 9 = 10 So, the third term is 10.
Calculating the Fourth Term To find the fourth term, a 4 , we substitute k = 4 into the expression 1 + 3 k :
a 4 = 1 + 3 ( 4 ) = 1 + 12 = 13 So, the fourth term is 13.
Calculating the Fifth Term To find the fifth term, a 5 , we substitute k = 5 into the expression 1 + 3 k :
a 5 = 1 + 3 ( 5 ) = 1 + 15 = 16 So, the fifth term is 16.
Final Answer Therefore, the first five terms of the arithmetic series are: a 1 = 4 a 2 = 7 a 3 = 10 a 4 = 13 a 5 = 16
Examples
Arithmetic series are used in various real-life applications, such as calculating the total cost of items that increase at a constant rate, predicting the number of seats in an auditorium where each row has a fixed number of additional seats, or determining the distance traveled by an object with constant acceleration. For example, if you save $100 in the first month and increase your savings by $50 each subsequent month, the total savings after a year can be calculated using an arithmetic series. Understanding arithmetic series helps in financial planning, engineering calculations, and many other practical scenarios.
The terms of the arithmetic series are a 1 = 4 , a 2 = 7 , a 3 = 10 , a 4 = 13 , and a 5 = 16 . These values are obtained by substituting the integers 1 to 5 into the expression 1 + 3 k . Therefore, the series consists of the following terms: 4, 7, 10, 13, and 16.
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