Evaluate [tex] \sum_{n=1}^{35} n^3+\sum_{n=1}^{35} n [/tex]
Answer (2)
We are asked to find the sum of ∑ n = 1 35 n 3 and ∑ n = 1 35 n .
Use the formula for the sum of the first N cubes: ∑ n = 1 N n 3 = ( 2 N ( N + 1 ) ) 2 with N=35.
Use the formula for the sum of the first N integers: ∑ n = 1 N n = 2 N ( N + 1 ) with N=35.
Add the results of the two sums to obtain the final answer: 397530 .
Explanation
Understanding the Problem We are asked to evaluate the sum of two series: ∑ n = 1 35 n 3 and ∑ n = 1 35 n . The first series is the sum of cubes from n = 1 to n = 35 , and the second series is the sum of integers from n = 1 to n = 35 . We will use the formulas for the sum of the first N cubes and the sum of the first N integers to solve this problem.
Stating the Formulas The formula for the sum of the first N cubes is given by: n = 1 ∑ N n 3 = ( 2 N ( N + 1 ) ) 2 The formula for the sum of the first N integers is given by: n = 1 ∑ N n = 2 N ( N + 1 ) In our case, N = 35 .
Calculating the Sums Now, we will substitute N = 35 into the formulas: n = 1 ∑ 35 n 3 = ( 2 35 ( 35 + 1 ) ) 2 = ( 2 35 × 36 ) 2 = ( 35 × 18 ) 2 = 63 0 2 = 396900 n = 1 ∑ 35 n = 2 35 ( 35 + 1 ) = 2 35 × 36 = 35 × 18 = 630
Adding the Results Finally, we add the results of the two sums to obtain the final answer: n = 1 ∑ 35 n 3 + n = 1 ∑ 35 n = 396900 + 630 = 397530 Therefore, the sum of the two series is 397530.
Final Answer The final answer is 397530.
Examples
Understanding series and summation formulas is crucial in many fields, such as physics and computer science. For example, when calculating the total energy of a system with discrete energy levels, you might use summation formulas to find the total energy. Similarly, in computer science, when analyzing the complexity of algorithms, summation formulas can help determine the total number of operations performed. These concepts are also used in financial mathematics to calculate the future value of annuities or investments with regular payments. Knowing how to efficiently compute sums allows for better modeling and prediction in these areas.
The evaluated expression ∑ n = 1 35 n 3 + ∑ n = 1 35 n results in 397530 by first calculating each sum separately using the appropriate formulas and then adding the results together.
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