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Question
Which of the following expressions correctly uses the properties of summations to represent sigma-summation Underscript lower i equals 1 overscript 18 EndScripts left parenthesis 7 lower I squared plus 9 right parenthesis?
(1 point)
Responses

7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i squared plus 9
Image with alt text: 7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i squared plus 9

7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts left parenthesis lower i squared plus 9 right parenthesis
Image with alt text: 7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts left parenthesis lower i squared plus 9 right parenthesis

7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i squared plus 18 dot 9
Image with alt text: 7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i squared plus 18 dot 9

7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i dot sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i plus 18 dot 9
Image with alt text: 7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i dot sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i plus 18 dot 9

1 answer

To represent the given summation \( \sum_{i=1}^{18} (7i^2 + 9) \) using the properties of summation, we can apply the linearity of the summation operator. The properties state that:

  1. \( \sum (a + b) = \sum a + \sum b \)
  2. \( \sum (c \cdot a) = c \cdot \sum a \)

Applying these properties to the expression:

\[ \sum_{i=1}^{18} (7i^2 + 9) = \sum_{i=1}^{18} (7i^2) + \sum_{i=1}^{18} (9) \]

Using property 2 for the first term:

\[ = 7 \sum_{i=1}^{18} i^2 + \sum_{i=1}^{18} 9 \]

The second term can be simplified further. Since there are 18 terms of 9:

\[ \sum_{i=1}^{18} 9 = 18 \cdot 9 \]

So, we can express the entire summation as:

\[ \sum_{i=1}^{18} (7i^2 + 9) = 7 \sum_{i=1}^{18} i^2 + 18 \cdot 9 \]

From the given response options, the one that matches this expression is:

7 \( \sum_{i=1}^{18} i^2 + 18 \cdot 9 \)

This corresponds to:

7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i squared plus 18 dot 9.

So the correct option is: 7 sigma-summation Underscript lower i equals 1 overscript 18 EndScripts lower i squared plus 18 dot 9.