Which summation properties or formulas must be used to evaluate this summation? Check all that apply.
[tex]$\sum_{n=1}^{35} n\left(n^2+1\right)$[/tex]
[tex]$\sum_{i=1}^n c=c n$[/tex]
[tex]$\sum_{i=1}^n i=\frac{n(n+1)}{2}$[/tex]
[tex]$\sum_{i=i_0}^n c a_i=c \sum_{i=i_0}^n a_i$[/tex]
[tex]$\sum_{i=1}^n i^2=\frac{n(n+1)(2 n+1)}{6}$[/tex]
[tex]$\sum_{i=i_0}^n\left(a_i \pm b_i\right)=\sum_{i=i_0}^n a_i \pm \sum_{1=1}^n b_i$[/tex]
[tex]$\sum_{i=1}^n i^3=\left[\frac{n(n+1)}{2}\right]^2$[/tex]
Answer (2)
Distribute n in the expression n ( n 2 + 1 ) to get n 3 + n .
Rewrite the summation as ∑ n = 1 35 ( n 3 + n ) .
Use the property ∑ i = i 0 n ( a i ± b i ) = ∑ i = i 0 n a i ± ∑ 1 = 1 n b i to separate the summation into ∑ n = 1 35 n 3 + ∑ n = 1 35 n .
Use the formulas ∑ i = 1 n i 3 = [ 2 n ( n + 1 ) ] 2 and ∑ i = 1 n i = 2 n ( n + 1 ) to evaluate the respective summations. The required summation properties are ∑ i = i 0 n ( a i ± b i ) = ∑ i = i 0 n a i ± ∑ 1 = 1 n b i , ∑ i = 1 n i = 2 n ( n + 1 ) , and ∑ i = 1 n i 3 = [ 2 n ( n + 1 ) ] 2 .
Explanation
Problem Analysis We are given the summation ∑ n = 1 35 n ( n 2 + 1 ) and asked to identify the summation properties or formulas needed to evaluate it.
Simplifying the Expression First, we simplify the expression inside the summation by distributing n : n ( n 2 + 1 ) = n 3 + n
Rewriting the Summation Now, we can rewrite the summation as: n = 1 ∑ 35 ( n 3 + n )
Splitting the Summation We can use the summation property that allows us to split a sum into separate summations: i = i 0 ∑ n ( a i ± b i ) = i = i 0 ∑ n a i ± i = i 0 ∑ n b i Applying this property, we get: n = 1 ∑ 35 ( n 3 + n ) = n = 1 ∑ 35 n 3 + n = 1 ∑ 35 n
Identifying Necessary Formulas We need the formulas for the sum of the first n cubes and the sum of the first n integers. These are given as: i = 1 ∑ n i 3 = [ 2 n ( n + 1 ) ] 2 i = 1 ∑ n i = 2 n ( n + 1 )
Conclusion Therefore, the summation properties and formulas needed to evaluate the given summation are:
∑ i = i 0 n ( a i ± b i ) = ∑ i = i 0 n a i ± ∑ 1 = 1 n b i
∑ i = 1 n i = 2 n ( n + 1 )
∑ i = 1 n i 3 = [ 2 n ( n + 1 ) ] 2
Examples
Summation formulas are used in many fields, such as physics, engineering, and computer science. For example, when calculating the total displacement of an object moving with variable velocity, we can use summation to add up the displacements over small time intervals. Similarly, in computer science, summations are used to analyze the complexity of algorithms and to calculate the total amount of work done by a program.
To evaluate the summation ∑ n = 1 35 n ( n 2 + 1 ) , we need to distribute and split the summation using basic properties. The required formulas include the sum of integers and the sum of cubes, which are used to calculate the final result. Thus, the overall summation evaluates to 397530.
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