What is the sum of the series below?
[tex]$\sum_{n=1}^{30}\left(n\left(n^2+4\right)\right)$[/tex]
Answer (2)
Rewrite the series as ∑ n = 1 30 ( n 3 + 4 n ) .
Separate the sum: ∑ n = 1 30 n 3 + 4 ∑ n = 1 30 n .
Apply the formulas for the sum of cubes and the sum of integers.
Calculate the final sum: 216225 + 1860 = 218085 .
Explanation
Understanding the Problem We are asked to find the sum of the series ∑ n = 1 30 n ( n 2 + 4 ) .
Rewriting the Series The series can be rewritten as ∑ n = 1 30 ( n 3 + 4 n ) .
Separating the Sum We need to evaluate the sum ∑ n = 1 30 ( n 3 + 4 n ) = ∑ n = 1 30 n 3 + 4 ∑ n = 1 30 n .
Sum of Cubes Formula We will use the formula for the sum of the first n cubes: ∑ i = 1 n i 3 = ( 2 n ( n + 1 ) ) 2 .
Sum of Integers Formula We will use the formula for the sum of the first n integers: ∑ i = 1 n i = 2 n ( n + 1 ) .
Substitution Substitute n = 30 into both formulas.
Calculating Sum of Cubes Calculate ∑ n = 1 30 n 3 = ( 2 30 ( 30 + 1 ) ) 2 = ( 2 30 ( 31 ) ) 2 = ( 15 × 31 ) 2 = 46 5 2 = 216225 .
Calculating Sum of Integers Calculate ∑ n = 1 30 n = 2 30 ( 30 + 1 ) = 2 30 ( 31 ) = 15 ( 31 ) = 465 .
Calculating 4 times Sum of Integers Calculate 4 ∑ n = 1 30 n = 4 ( 465 ) = 1860 .
Final Calculation Calculate ∑ n = 1 30 n 3 + 4 ∑ n = 1 30 n = 216225 + 1860 = 218085 .
Conclusion Therefore, the sum of the series is 218085.
Examples
Understanding series and their sums is crucial in many fields, such as physics and engineering. For example, when calculating the total energy of a system that changes over discrete time intervals, we often use series to model the cumulative effect. Imagine designing a bridge where each component adds a certain amount of stress. The total stress can be modeled as a series, and finding the sum helps engineers ensure the bridge's stability and safety. This problem demonstrates how to efficiently compute such sums using mathematical formulas, which is essential for accurate and reliable predictions in real-world applications.
The sum of the series ∑ n = 1 30 n ( n 2 + 4 ) can be calculated by rewriting it as ∑ n = 1 30 ( n 3 + 4 n ) and separating it into two sums. Using the formulas for the sum of cubes and the sum of integers, we derive the final result of 218085 .
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