Which expression is equivalent to $\sum_{n=1}^{60}(2 n-1)^2 $? Check all that apply.

$\sum_{n=1}^{60} 4 n^2-4 n-1$

$4 \sum_{n=1}^{\infty} n^2-4 \sum_{n=1}^{\infty} n+\sum_{n=1}^{\infty} 1$

$4 \sum_{n=1}^{\infty} n^2-4 \sum_{n=1}^{\infty} n-\sum_{n=1}^{\infty} 1$

$4 \sum_{n=1}^{\infty} n^2-4 \sum_{n=1}^{\infty} n+60$

Expand the original expression: ∑ n = 1 60 ​ ( 2 n − 1 ) 2 = ∑ n = 1 60 ​ ( 4 n 2 − 4 n + 1 ) .
Rewrite the sum: ∑ n = 1 60 ​ ( 4 n 2 − 4 n + 1 ) = 4 ∑ n = 1 60 ​ n 2 − 4 ∑ n = 1 60 ​ n + ∑ n = 1 60 ​ 1 .
Simplify the constant term: ∑ n = 1 60 ​ 1 = 60 .
The equivalent expression is: 4 ∑ n = 1 60 ​ n 2 − 4 ∑ n = 1 60 ​ n + 60 .

Explanation

Understanding the Problem We are given the expression ∑ n = 1 60 ​ ( 2 n − 1 ) 2 and we need to find equivalent expressions from the list.

Expanding the Expression First, let's expand the term inside the summation: ( 2 n − 1 ) 2 = 4 n 2 − 4 n + 1 . So, the original expression can be written as ∑ n = 1 60 ​ ( 4 n 2 − 4 n + 1 ) .

Analyzing the Options Now, let's analyze the given options:


Option 1: ∑ n = 1 60 ​ 4 n 2 − 4 n − 1 . This can be written as ∑ n = 1 60 ​ ( 4 n 2 − 4 n − 1 ) . Comparing this with ∑ n = 1 60 ​ ( 4 n 2 − 4 n + 1 ) , we see that they are not equal.
Option 2: 4 ∑ n = 1 ∞ ​ n 2 − 4 ∑ n = 1 ∞ ​ n + ∑ n = 1 ∞ ​ 1 . This involves summation to infinity, while our original expression is a finite sum up to 60. So, this option is not equivalent.
Option 3: 4 ∑ n = 1 ∞ ​ n 2 − 4 ∑ n = 1 ∞ ​ n − ∑ n = 1 ∞ ​ 1 . Similar to option 2, this also involves summation to infinity, so it's not equivalent.
Option 4: 4 ∑ n = 1 60 ​ n 2 − 4 ∑ n = 1 60 ​ n + 60 . Let's rewrite our original expression: ∑ n = 1 60 ​ ( 4 n 2 − 4 n + 1 ) = ∑ n = 1 60 ​ 4 n 2 − ∑ n = 1 60 ​ 4 n + ∑ n = 1 60 ​ 1 = 4 ∑ n = 1 60 ​ n 2 − 4 ∑ n = 1 60 ​ n + ∑ n = 1 60 ​ 1 . Since ∑ n = 1 60 ​ 1 = 60 , we have 4 ∑ n = 1 60 ​ n 2 − 4 ∑ n = 1 60 ​ n + 60 . This is exactly the same as option 4.

Final Answer Therefore, the equivalent expression is 4 ∑ n = 1 60 ​ n 2 − 4 ∑ n = 1 60 ​ n + 60 .

Examples
Understanding series and summations is crucial in many fields, such as physics and engineering. For example, when calculating the total energy of a system with multiple components, you might use a summation to add up the energy of each component. Similarly, in finance, you might use summations to calculate the total return on an investment over a period of time. The ability to manipulate and simplify these expressions can greatly aid in these calculations.

Answered by GinnyAnswer | 2025-07-03

The equivalent expression for ∑ n = 1 60 ​ ( 2 n − 1 ) 2 is 4 ∑ n = 1 60 ​ n 2 − 4 ∑ n = 1 60 ​ n + 60 , which matches option 4 in the list provided.
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Answered by Anonymous | 2025-07-04