Which of the following shows one way to simplify $\sum_{n=1}^{43}(3+9 n) ?$

A. $\sum_{n=1}^{43} 3+\sum_{n=1}^{43} 9 n$
B. $\sum_{n=1}^{43} 3+\sum_{n=1}^{43} 9+\sum_{n=1}^{43} n$
C. $3 \sum_{n=1}^{43} 9 n$

The summation of a sum is the sum of the summations: ∑ n = 1 43 ​ ( 3 + 9 n ) = ∑ n = 1 43 ​ 3 + ∑ n = 1 43 ​ 9 n .
A constant factor can be taken out of the summation: ∑ n = 1 43 ​ 9 n = 9 ∑ n = 1 43 ​ n .
Therefore, ∑ n = 1 43 ​ ( 3 + 9 n ) = ∑ n = 1 43 ​ 3 + 9 ∑ n = 1 43 ​ n .
The correct simplification is n = 1 ∑ 43 ​ 3 + n = 1 ∑ 43 ​ 9 n ​ .

Explanation

Understanding the Problem We are given the summation ∑ n = 1 43 ​ ( 3 + 9 n ) and asked to simplify it.

Splitting the Summation We can use the property that the summation of a sum is the sum of the summations: n = 1 ∑ 43 ​ ( 3 + 9 n ) = n = 1 ∑ 43 ​ 3 + n = 1 ∑ 43 ​ 9 n

Factoring out the Constant We can also use the property that a constant factor can be taken out of the summation: n = 1 ∑ 43 ​ 9 n = 9 n = 1 ∑ 43 ​ n

Final Simplification Therefore, we have: n = 1 ∑ 43 ​ ( 3 + 9 n ) = n = 1 ∑ 43 ​ 3 + 9 n = 1 ∑ 43 ​ n Comparing this with the given options, we see that the correct simplification is ∑ n = 1 43 ​ 3 + ∑ n = 1 43 ​ 9 n .


Examples
Understanding summations is crucial in many fields, such as physics and computer science. For instance, when calculating the total distance traveled by an object with varying speeds at different time intervals, you use a summation to add up the distances covered in each interval. Similarly, in computer science, summations are used to analyze the complexity of algorithms, where you might sum up the number of operations performed in each step of the algorithm to determine its overall efficiency. This concept is also used in finance to calculate the total return on investment over a period of time.

Answered by GinnyAnswer | 2025-07-03

The summation ∑ n = 1 43 ​ ( 3 + 9 n ) can be simplified using the property that the sum of a sum is the sum of the summations, resulting in ∑ n = 1 43 ​ 3 + ∑ n = 1 43 ​ 9 n . Therefore, the correct option is A. This means we separate the constant part and the variable part of the summation for evaluation.
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Answered by Anonymous | 2025-07-04