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:
- \( \sum (a + b) = \sum a + \sum b \)
- \( \sum (c \cdot a) = c \cdot \sum a \)
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.