What is the expanded form of the series represented below?
[tex]S_5=\sum_{k=1}^5[-3+(k-1) 5][/tex]
A. [tex]-3+2+7+12+17[/tex]
B. [tex]-3+3+8+13+18[/tex]
C. [tex]2+7+12+17+22[/tex]
D. [tex]5+2+(-1)+(-4)+(-7)[/tex]
Answer (2)
The problem asks for the expanded form of the series S 5 = ∑ k = 1 5 [ − 3 + ( k − 1 ) 5 ] .
We substitute k = 1 , 2 , 3 , 4 , 5 into the expression − 3 + ( k − 1 ) 5 .
The terms are calculated as − 3 , 2 , 7 , 12 , 17 .
The expanded form of the series is − 3 + 2 + 7 + 12 + 17 .
Explanation
Understanding the Problem We are asked to find the expanded form of the series S 5 = ∑ k = 1 5 [ − 3 + ( k − 1 ) 5 ] . This means we need to substitute k = 1 , 2 , 3 , 4 , 5 into the expression − 3 + ( k − 1 ) 5 and write out the terms.
Calculating the first term For k = 1 , the term is − 3 + ( 1 − 1 ) 5 = − 3 + 0 = − 3 .
Calculating the second term For k = 2 , the term is − 3 + ( 2 − 1 ) 5 = − 3 + 5 = 2 .
Calculating the third term For k = 3 , the term is − 3 + ( 3 − 1 ) 5 = − 3 + 10 = 7 .
Calculating the fourth term For k = 4 , the term is − 3 + ( 4 − 1 ) 5 = − 3 + 15 = 12 .
Calculating the fifth term For k = 5 , the term is − 3 + ( 5 − 1 ) 5 = − 3 + 20 = 17 .
Final Answer Therefore, the expanded form of the series is − 3 + 2 + 7 + 12 + 17 .
Examples
Series are used in many areas of mathematics, physics, and engineering. For example, they can be used to approximate the values of functions, to solve differential equations, and to model physical phenomena. In finance, series can be used to calculate the present value of a stream of future cash flows. Understanding series helps in modeling and predicting outcomes in various fields.
The expanded form of the series S 5 = ∑ k = 1 5 [ − 3 + ( k − 1 ) 5 ] is − 3 + 2 + 7 + 12 + 17 , which corresponds to choice A.
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