Which summation is equal to the following?
[tex]$\sum_{n=37}^{75} 3 n^2$[/tex]
A. [tex]$3 \sum_{n=37}^{75} n^2$[/tex]
B. [tex]$3 n \sum_{n=37}^{75} n$[/tex]
C. [tex]$\sum_{n=37}^{75} 3+\sum_{n=37}^{75} n^2$[/tex]
D. [tex]$\sum_{n=37}^{75} 3 \times \sum_{n=37}^{75} n^2$[/tex]
Answer (2)
Recognize that the constant factor 3 can be factored out of the summation.
Apply the property ∑ n = a b c f ( n ) = c ∑ n = a b f ( n ) .
Rewrite the given summation as 3 ∑ n = 37 75 n 2 .
Conclude that the equivalent summation is 3 n = 37 ∑ 75 n 2 .
Explanation
Understanding the Problem We are given the summation ∑ n = 37 75 3 n 2 and asked to find an equivalent expression from the given options.
Using the Constant Factor Property The key property to use here is that a constant factor can be moved outside the summation. That is, ∑ n = a b c f ( n ) = c ∑ n = a b f ( n ) , where c is a constant.
Applying the Property Applying this property to the given summation, we have n = 37 ∑ 75 3 n 2 = 3 n = 37 ∑ 75 n 2 .
Identifying the Correct Option Comparing this result with the given options, we see that the first option, 3 ∑ n = 37 75 n 2 , matches our result.
Final Answer Therefore, the summation equal to ∑ n = 37 75 3 n 2 is 3 ∑ n = 37 75 n 2 .
Examples
Summations are used in many areas of math and science. For example, in physics, you might use a summation to calculate the total energy of a system, where each term in the sum represents the energy of a single particle. In finance, summations are used to calculate the total return on an investment over a period of time. In computer science, summations are used to analyze the running time of algorithms. For instance, if an algorithm performs a certain operation f ( n ) for each input size n from 1 to N , the total number of operations can be expressed as ∑ n = 1 N f ( n ) .
The summation ∑ n = 37 75 3 n 2 can be rewritten by factoring out the constant 3, resulting in 3 ∑ n = 37 75 n 2 . Therefore, the correct answer is option A. This demonstrates how properties of summation can simplify expressions effectively.
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