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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

Answer 1

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.**