Which expression is equivalent to
$\sum_{n=1}^{35} n\left(n^2+1\right)$?

$\sum_{n=1}^{35} n \times \sum_{n=1}^{35}\left(n^2+1\right)$

$\sum_{n=1}^{35} n^3 \times \sum_{n=1}^{35} n$

$\sum_{n=1}^{35} n^3+\sum_{n=1}^{35} n$

Expand the original summation: ∑ n = 1 35 ​ n ( n 2 + 1 ) = ∑ n = 1 35 ​ ( n 3 + n ) .
Separate the summation into two parts: ∑ n = 1 35 ​ ( n 3 + n ) = ∑ n = 1 35 ​ n 3 + ∑ n = 1 35 ​ n .
Compare the result with the given expressions.
The equivalent expression is: n = 1 ∑ 35 ​ n 3 + n = 1 ∑ 35 ​ n ​

Explanation

Problem Analysis We are given the summation ∑ n = 1 35 ​ n ( n 2 + 1 ) and four expressions. Our goal is to identify which of the four expressions is equivalent to the given summation.

Expanding the Summation Let's expand the given summation: n = 1 ∑ 35 ​ n ( n 2 + 1 ) = n = 1 ∑ 35 ​ ( n 3 + n ) Now, we can use the property of summation to separate the terms: n = 1 ∑ 35 ​ ( n 3 + n ) = n = 1 ∑ 35 ​ n 3 + n = 1 ∑ 35 ​ n

Comparing with Given Expressions Now, let's compare this result with the four given expressions:

∑ n = 1 35 ​ n × ∑ n = 1 35 ​ ( n 2 + 1 ) : This is not equivalent because it's a product of two summations, not a sum.

∑ n = 1 35 ​ n 3 × ∑ n = 1 35 ​ n : This is also not equivalent because it's a product of two summations.

∑ n = 1 35 ​ n 3 + ∑ n = 1 35 ​ n : This is exactly what we derived.

Final Answer Therefore, the expression equivalent to ∑ n = 1 35 ​ n ( n 2 + 1 ) is ∑ n = 1 35 ​ n 3 + ∑ n = 1 35 ​ n .


Examples
Understanding summations is crucial in many fields, such as physics and computer science. For instance, when calculating the total energy of a system with multiple particles, you might use a summation to add up the kinetic energy 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 to determine its overall efficiency. This problem demonstrates a basic manipulation of summations, which is a fundamental skill in these areas.

Answered by GinnyAnswer | 2025-07-03

The expression equivalent to ∑ n = 1 35 ​ n ( n 2 + 1 ) is ∑ n = 1 35 ​ n 3 + ∑ n = 1 35 ​ n . This was determined by expanding the original summation and comparing it with the provided options. The correct choice from the options given is the third one, which is ∑ n = 1 35 ​ n 3 + ∑ n = 1 35 ​ n .
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Answered by Anonymous | 2025-07-04