The sum $\sum_{n=1}^{50} n(n+6)=$
Answer (2)
Expand the summand: n ( n + 6 ) = n 2 + 6 n .
Split the sum: ∑ n = 1 50 n ( n + 6 ) = ∑ n = 1 50 n 2 + 6 ∑ n = 1 50 n .
Apply the formulas for the sum of first n integers and squares.
Calculate the final result: ∑ n = 1 50 n ( n + 6 ) = 50575 . The final answer is 50575 .
Explanation
Understanding the Problem We are asked to find the sum of the series ∑ n = 1 50 n ( n + 6 ) . This means we need to add up the values of n ( n + 6 ) for each integer n from 1 to 50.
Expanding the Expression First, let's expand the expression inside the summation: n ( n + 6 ) = n 2 + 6 n . So, we have ∑ n = 1 50 ( n 2 + 6 n ) .
Splitting the Summation Now, we can split the summation into two separate summations: ∑ n = 1 50 ( n 2 + 6 n ) = ∑ n = 1 50 n 2 + ∑ n = 1 50 6 n = ∑ n = 1 50 n 2 + 6 ∑ n = 1 50 n .
Using Known Formulas We know the formulas for the sum of the first N integers and the sum of the first N squares: n = 1 ∑ N n = 2 N ( N + 1 ) n = 1 ∑ N n 2 = 6 N ( N + 1 ) ( 2 N + 1 ) In our case, N = 50 .
Calculating the Sums Let's calculate ∑ n = 1 50 n :
n = 1 ∑ 50 n = 2 50 ( 50 + 1 ) = 2 50 ( 51 ) = 25 ( 51 ) = 1275 And let's calculate ∑ n = 1 50 n 2 :
n = 1 ∑ 50 n 2 = 6 50 ( 50 + 1 ) ( 2 ( 50 ) + 1 ) = 6 50 ( 51 ) ( 101 ) = 1 25 ( 17 ) ( 101 ) = 42925
Final Calculation Now, substitute these values back into our expression: n = 1 ∑ 50 n 2 + 6 n = 1 ∑ 50 n = 42925 + 6 ( 1275 ) = 42925 + 7650 = 50575
Final Answer Therefore, the sum ∑ n = 1 50 n ( n + 6 ) = 50575 .
Examples
Understanding series and summations is crucial in many fields, such as physics and computer science. For example, when calculating the total distance traveled by an object with increasing acceleration, you might use a summation similar to this one. In finance, understanding series can help calculate the future value of an investment with regular contributions. These mathematical tools provide a foundation for modeling and solving real-world problems.
The sum ∑ n = 1 50 n ( n + 6 ) is calculated by expanding and splitting the summand into two parts, applying known formulas for sums of integers and squares. After performing the calculations, we find the result to be 50575 .
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