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=4 \\
a_2=7 \\
a_3=10 \\
a_4=13 \\
a_5=16
\end{array}[/tex]
What is the value of [tex]S_5[/tex]?
[tex]S_5=16[/tex]
Answer (1)
The terms of the arithmetic series are a 1 = 4 , a 2 = 7 , a 3 = 10 , a 4 = 13 , a 5 = 16 .
Calculate the sum of the series: S 5 = 4 + 7 + 10 + 13 + 16 .
The sum is S 5 = 50 .
Therefore, the value of S 5 is 50 .
Explanation
Understanding the Problem We are given an arithmetic series S 5 = ∑ k = 1 5 ( 1 + 3 k ) . We need to find the terms of the series and the value of S 5 . The terms are already given as a 1 = 4 , a 2 = 7 , a 3 = 10 , a 4 = 13 , a 5 = 16 . However, the value of S 5 is incorrectly given as 16. We need to calculate the correct value of S 5 .
Calculating the Sum To find the value of S 5 , we can sum the terms of the series: S 5 = a 1 + a 2 + a 3 + a 4 + a 5 = 4 + 7 + 10 + 13 + 16 .
Finding the Value of S_5 Adding the terms, we get: S 5 = 4 + 7 + 10 + 13 + 16 = 50 Alternatively, we can use the formula for the sum of an arithmetic series: S n = 2 n ( a 1 + a n ) , where n = 5 , a 1 = 4 , and a 5 = 16 .
S 5 = 2 5 ( 4 + 16 ) = 2 5 ( 20 ) = 5 × 10 = 50 Thus, the value of S 5 is 50.
Final Answer Therefore, the terms of the arithmetic series are a 1 = 4 , a 2 = 7 , a 3 = 10 , a 4 = 13 , a 5 = 16 , and the value of S 5 is 50.
Examples
Arithmetic series are useful in many real-life situations. For example, consider a stack of logs where each layer has one fewer log than the layer below it. If the bottom layer has 10 logs, the next layer has 9, and so on, until the top layer has 1 log, the total number of logs can be calculated using an arithmetic series. This concept is also applicable in calculating the total cost of items when the price increases linearly over time, or in determining the total distance traveled when the speed increases at a constant rate.
