Which expression is equal to $\sum_{n=1}^{50} n^2+6 \sum_{n=1}^{50} n$?
A. $\frac{50(51)(52)}{6}+6\left(\frac{(50)(51)}{2}\right)$
B. $\frac{50(51)(101)}{6}+\left(\frac{(50)(51)}{2}\right)$
C. $\frac{50(51)(101)}{6}+6\left(\frac{(50)(51)}{2}\right)$
Answer (2)
Calculate the sum of the first 50 squares: ∑ n = 1 50 n 2 = 6 50 ( 51 ) ( 101 ) .
Calculate the sum of the first 50 integers: ∑ n = 1 50 n = 2 50 ( 51 ) .
Substitute these values into the original expression: 6 50 ( 51 ) ( 101 ) + 6 ( 2 50 ( 51 ) ) .
The equivalent expression is 6 50 ( 51 ) ( 101 ) + 6 ( 2 ( 50 ) ( 51 ) ) .
Explanation
Understanding the Problem We are given the expression ∑ n = 1 50 n 2 + 6 ∑ n = 1 50 n and asked to find an equivalent expression from the given options. We will use the formulas for the sum of the first n squares and the sum of the first n integers to simplify the expression and then compare it to the given options.
Sum of Squares The formula for the sum of the first n squares is ∑ i = 1 n i 2 = 6 n ( n + 1 ) ( 2 n + 1 ) . For n = 50 , this is n = 1 ∑ 50 n 2 = 6 50 ( 50 + 1 ) ( 2 ( 50 ) + 1 ) = 6 50 ( 51 ) ( 101 ) .
Sum of Integers The formula for the sum of the first n integers is ∑ i = 1 n i = 2 n ( n + 1 ) . For n = 50 , this is n = 1 ∑ 50 n = 2 50 ( 50 + 1 ) = 2 50 ( 51 ) .
Combining the Results Now we substitute these values into the original expression: n = 1 ∑ 50 n 2 + 6 n = 1 ∑ 50 n = 6 50 ( 51 ) ( 101 ) + 6 ( 2 50 ( 51 ) ) . Comparing this to the given options, we see that it matches the third option.
Final Answer Therefore, the expression equal to ∑ n = 1 50 n 2 + 6 ∑ n = 1 50 n is 6 50 ( 51 ) ( 101 ) + 6 ( 2 ( 50 ) ( 51 ) ) .
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 time complexity of algorithms, you often encounter summations that need to be simplified to understand how the algorithm scales with input size. Knowing these formulas allows for efficient calculation and analysis in these scenarios.
The expression ∑ n = 1 50 n 2 + 6 ∑ n = 1 50 n simplifies to 6 50 ( 51 ) ( 101 ) + 6 ( 2 ( 50 ) ( 51 ) ) . This matches option C, making it the correct answer. Therefore, the answer is C.
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