Which expression is equivalent to [tex]$\sum_{n=10}^{15}\left(n+\frac{3}{n}\right)$[/tex] ?
A. [tex]$\sum_{n=10}^{15} n+\sum_{n=10}^{15} \frac{3}{n}$[/tex]
B. [tex]$\sum_{n=10}^{15} n+\sum_{n=10}^{15} \div \sum_{n=10}^{15} n$[/tex]
C. [tex]$n \sum_{n=10}^{15} \frac{3}{n}$[/tex]
D. [tex]$\frac{3}{n} \sum_{n=10}^{15} n$[/tex]
Answer (2)
Apply the summation property: ∑ i = a b ( x i + y i ) = ∑ i = a b x i + ∑ i = a b y i .
Apply this property to the given expression: ∑ n = 10 15 ( n + n 3 ) = ∑ n = 10 15 n + ∑ n = 10 15 n 3 .
Compare the resulting expression with the options provided.
The equivalent expression is: n = 10 ∑ 15 n + n = 10 ∑ 15 n 3
Explanation
Understanding the Problem We are given the summation ∑ n = 10 15 ( n + n 3 ) and we want to find an equivalent expression from the options provided.
Key Property of Summations The key property of summations that we'll use is that the summation of a sum is the sum of the summations. In other words, ∑ i = a b ( x i + y i ) = ∑ i = a b x i + ∑ i = a b y i .
Applying the Property Applying this property to the given expression, we have: n = 10 ∑ 15 ( n + n 3 ) = n = 10 ∑ 15 n + n = 10 ∑ 15 n 3
Comparing with Options Now we compare this result with the options provided. The first option is ∑ n = 10 15 n + ∑ n = 10 15 n 3 , which matches our result.
Final Answer Therefore, the expression equivalent to ∑ n = 10 15 ( n + n 3 ) is ∑ n = 10 15 n + ∑ n = 10 15 n 3 .
Examples
Understanding summations is crucial in many fields, such as physics and computer science. For example, when calculating the total energy of a system with multiple particles, you might sum the kinetic and potential energies of each particle. Similarly, in computer science, summations are used to analyze the complexity of algorithms, where you might sum the number of operations performed in each step of the algorithm. This problem demonstrates a basic property of summations that simplifies complex expressions into manageable parts.
The summation ∑ n = 10 15 ( n + n 3 ) can be split into two separate summations using the property of summation. The correct equivalent expression is option A: ∑ n = 10 15 n + ∑ n = 10 15 n 3 .
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