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]
Which is equivalent to the following?
D. [tex]$3 \sum_{n=37}^{75} n^2$[/tex]
E. [tex]$3\left[\sum_{n=1}^{75} n^2-\sum_{n=1}^{37} n^2\right]$[/tex]
F. [tex]$3\left[\sum_{n=37}^{75} n^2-\sum_{n=1}^{36} n^2\right]$[/tex]
G. [tex]$3\left[\sum_{n=1}^{75} n^2\right]-\sum_{n=1}^{36} n^2$[/tex]
H. [tex]$3\left[\sum_{n=1}^{75} n^2-\sum_{n=1}^{36} n^2\right]$[/tex]
Answer (2)
The summation ∑ n = 37 75 3 n 2 is equivalent to 3 ∑ n = 37 75 n 2 .
The expression 3 ∑ n = 37 75 n 2 is equivalent to 3 [ ∑ n = 1 75 n 2 − ∑ n = 1 36 n 2 ] .
Therefore, the final answers are 3 ∑ n = 37 75 n 2 and 3 [ ∑ n = 1 75 n 2 − ∑ n = 1 36 n 2 ] .
Explanation
Analyzing the First Summation We are given the summation ∑ n = 37 75 3 n 2 and asked to find an equivalent expression from the provided options. The key property to remember is that a constant factor can be moved outside of a summation. That is, ∑ n = a b c f ( n ) = c ∑ n = a b f ( n ) .
Finding the Equivalent Expression Applying this property to our given summation, we have: n = 37 ∑ 75 3 n 2 = 3 n = 37 ∑ 75 n 2 Therefore, the first equivalent expression is 3 ∑ n = 37 75 n 2 .
Analyzing the Second Expression Now, we are given the expression 3 ∑ n = 37 75 n 2 and asked to find an equivalent expression from the provided options. We can rewrite the summation ∑ n = 37 75 n 2 as the difference of two summations: n = 37 ∑ 75 n 2 = n = 1 ∑ 75 n 2 − n = 1 ∑ 36 n 2 This is because the summation from n = 1 to 75 includes all terms from n = 1 to 36 and from n = 37 to 75 . So, subtracting the summation from n = 1 to 36 leaves us with the summation from n = 37 to 75 .
Finding the Equivalent Expression Multiplying both sides by 3, we get: 3 n = 37 ∑ 75 n 2 = 3 [ n = 1 ∑ 75 n 2 − n = 1 ∑ 36 n 2 ] Therefore, the equivalent expression is 3 [ ∑ n = 1 75 n 2 − ∑ n = 1 36 n 2 ] .
Examples
Summations are used in many areas of mathematics and computer science. For example, in physics, you might use a summation to calculate the total energy of a system consisting of many particles, where each particle has a different energy level. In finance, you might use a summation to calculate the total return on an investment portfolio over a period of time, where each investment has a different return. In computer science, summations are used in algorithms to calculate the total number of operations required to process a dataset.
The summation n = 37 ∑ 75 3 n 2 is equivalent to 3 n = 37 ∑ 75 n 2 , found in option A. Additionally, it can also be expressed as 3 [ n = 1 ∑ 75 n 2 − n = 1 ∑ 36 n 2 ] , which corresponds to option H.
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