Which of the following represents the value of the series below?
[tex]S_{28}=\sum_{k=1}^{28}(9 k-4)[/tex]
A. [tex]S_{28}=28\left(\frac{5+248}{2}\right)[/tex]
B. [tex]S_{28}=(9(28)-4)+(9(1)-4)[/tex]
C. [tex]S_{28}=28(5+248)[/tex]
D. [tex]S_{28}=28\left(\frac{248-5}{2}\right)[/tex]
Answer (2)
Calculate the sum of the series: S 28 = ∑ k = 1 28 ( 9 k − 4 ) = 9 ∑ k = 1 28 k − ∑ k = 1 28 4 .
Use the formula for the sum of the first n integers: ∑ k = 1 28 k = 2 28 ( 29 ) = 406 .
Compute the sum: S 28 = 9 ( 406 ) − 4 ( 28 ) = 3654 − 112 = 3542 .
Compare the result with the given options and identify the correct one: S 28 = 28 ( 2 5 + 248 ) .
Explanation
Problem Analysis We are given a series S 28 = ∑ k = 1 28 ( 9 k − 4 ) and several options for its value. Our goal is to determine which of the provided options correctly represents the sum of the series.
Rewriting the Series First, let's compute the sum of the series directly. We can rewrite the series as follows: S 28 = k = 1 ∑ 28 ( 9 k − 4 ) = 9 k = 1 ∑ 28 k − k = 1 ∑ 28 4
Sum of First 28 Integers We know that the sum of the first n integers is given by the formula ∑ k = 1 n k = 2 n ( n + 1 ) . In our case, n = 28 , so k = 1 ∑ 28 k = 2 28 ( 28 + 1 ) = 2 28 × 29 = 14 × 29 = 406
Calculating the Series Sum Also, ∑ k = 1 28 4 = 4 × 28 = 112 . Therefore, S 28 = 9 × 406 − 112 = 3654 − 112 = 3542
Evaluating the Options Now, let's evaluate the given options:
Option 1: S 28 = 28 ( 2 5 + 248 ) = 28 ( 2 253 ) = 14 × 253 = 3542
Option 2: S 28 = ( 9 ( 28 ) − 4 ) + ( 9 ( 1 ) − 4 ) = ( 252 − 4 ) + ( 9 − 4 ) = 248 + 5 = 253
Option 3: S 28 = 28 ( 5 + 248 ) = 28 ( 253 ) = 7084
Option 4: S 28 = 28 ( 2 248 − 5 ) = 28 ( 2 243 ) = 14 × 243 = 3402
Finding the Correct Option Comparing the calculated value of the series S 28 = 3542 with the values of the options, we find that Option 1 matches the calculated value.
Final Answer Therefore, the correct representation of the series is S 28 = 28 ( 2 5 + 248 ) .
Examples
Consider a scenario where you're calculating the total cost of items purchased over 28 days, with the cost increasing linearly each day. If the cost on day k is given by 9 k − 4 , then the series S 28 = ∑ k = 1 28 ( 9 k − 4 ) represents the total cost over the 28 days. Being able to evaluate such a series efficiently helps in financial planning and budgeting, allowing you to quickly determine the overall expenditure.
The sum of the series S 28 = ∑ k = 1 28 ( 9 k − 4 ) is calculated to be 3542. The correct representation of this sum from the given options is Option A: S 28 = 28 ( 2 5 + 248 ) . Thus, the answer is Option A.
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