Evaluate [tex] \sum_{n=1}^{35} n^3+\sum_{n=1}^{35} n [/tex]
Answer (2)
Recall the formula for the sum of the first n cubes: ∑ k = 1 n k 3 = ( 2 n ( n + 1 ) ) 2 .
Recall the formula for the sum of the first n integers: ∑ k = 1 n k = 2 n ( n + 1 ) .
Calculate the sum of the first 35 cubes: ∑ n = 1 35 n 3 = 396900 .
Calculate the sum of the first 35 integers: ∑ n = 1 35 n = 630 .
Add the two sums to get the final answer: 397530 .
Explanation
Understanding the Problem We are asked to evaluate the sum of two series. The first series is the sum of cubes from n=1 to 35, and the second series is the sum of integers from n=1 to 35.
Defining the Sum We need to find the value of n = 1 ∑ 35 n 3 + n = 1 ∑ 35 n
Sum of Cubes Formula Recall the formula for the sum of the first n cubes: k = 1 ∑ n k 3 = ( 2 n ( n + 1 ) ) 2
Sum of Integers Formula Recall the formula for the sum of the first n integers: k = 1 ∑ n k = 2 n ( n + 1 )
Calculating Sum of Cubes Apply the formula for the sum of the first n cubes with n=35: n = 1 ∑ 35 n 3 = ( 2 35 ( 35 + 1 ) ) 2 = ( 2 35 ⋅ 36 ) 2 = ( 35 ⋅ 18 ) 2 = 63 0 2 = 396900
Calculating Sum of Integers Apply the formula for the sum of the first n integers with n=35: n = 1 ∑ 35 n = 2 35 ( 35 + 1 ) = 2 35 ⋅ 36 = 35 ⋅ 18 = 630
Adding the Sums Add the two sums: n = 1 ∑ 35 n 3 + n = 1 ∑ 35 n = 396900 + 630 = 397530
Final Answer Therefore, the final answer is 397530.
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 increases with the cube of a variable (like velocity) and also has a linear component, you would use these formulas. Imagine designing a rocket where the fuel consumption increases cubically with speed, but also has a base consumption rate. Knowing how to sum these series helps engineers predict total fuel needs for different mission profiles, ensuring efficient and safe operation.
The evaluation of ∑ n = 1 35 n 3 + ∑ n = 1 35 n yields 397530 . This is calculated using relevant formulas for the sums of cubes and integers. Finally, by adding these two sums, we arrive at the answer.
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