Describe the steps to evaluate the summation. What is the sum?
[tex] \sum_{j=1}^{200} 2 j(j+3) [/tex]
Answer (2)
Expand the summand: 2 j ( j + 3 ) = 2 j 2 + 6 j .
Apply the summation: ∑ j = 1 200 ( 2 j 2 + 6 j ) = 2 ∑ j = 1 200 j 2 + 6 ∑ j = 1 200 j .
Use the summation formulas: ∑ j = 1 n j = 2 n ( n + 1 ) and ∑ j = 1 n j 2 = 6 n ( n + 1 ) ( 2 n + 1 ) .
Calculate the final result: The sum is 5494000 .
Explanation
Understanding the Problem We are asked to evaluate the summation ∑ j = 1 200 2 j ( j + 3 ) . This means we need to find the sum of the expression 2 j ( j + 3 ) as j takes on integer values from 1 to 200.
Expanding the Summand First, let's expand the summand: 2 j ( j + 3 ) = 2 j 2 + 6 j . This will make it easier to apply the summation.
Applying the Summation Now, we can apply the summation to each term: ∑ j = 1 200 ( 2 j 2 + 6 j ) = 2 ∑ j = 1 200 j 2 + 6 ∑ j = 1 200 j . We can use the formulas for the sum of the first n integers and the sum of the first n squares to evaluate these summations.
Recalling Summation Formulas Recall the formulas for the sum of the first n integers and the sum of the first n squares: j = 1 ∑ n j = 2 n ( n + 1 ) j = 1 ∑ n j 2 = 6 n ( n + 1 ) ( 2 n + 1 ) We will use these formulas with n = 200 .
Calculating Sums Substitute n = 200 into the formulas: j = 1 ∑ 200 j = 2 200 ( 200 + 1 ) = 2 200 ( 201 ) = 100 ( 201 ) = 20100 j = 1 ∑ 200 j 2 = 6 200 ( 200 + 1 ) ( 2 ( 200 ) + 1 ) = 6 200 ( 201 ) ( 401 ) = 6 16120200 = 2686700
Final Calculation Now, substitute these results back into the expression:
2 ∑ j = 1 200 j 2 + 6 ∑ j = 1 200 j = 2 ( 2686700 ) + 6 ( 20100 ) = 5373400 + 120600 = 5494000
Conclusion Therefore, the sum of the given expression is 5,494,000.
Examples
Summations are used extensively in computer science, particularly in the analysis of algorithms. For example, if you want to determine the total number of operations an algorithm performs, you might express it as a summation. Understanding how to evaluate summations allows you to predict the performance of your code and optimize it for efficiency. In finance, summations are used to calculate the future value of a series of cash flows. For example, if you invest a fixed amount of money each month, you can use a summation to calculate the total value of your investment after a certain period.
To evaluate the summation ∑ j = 1 200 2 j ( j + 3 ) , expand to get 2 j 2 + 6 j , apply the summation formulas, and calculate to find the final sum is 5 , 494 , 000 .
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