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(D) We know that for any triangle $ABC$ with centroid $O$,the vector sum $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overright

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$3$

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(D) We know that for any triangle $ABC$ with centroid $O$,the vector sum $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{0}$.We can express the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ in terms of the position vectors relative to the centroid $O$:$\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = \overrightarrow{AO} + \overrightarrow{OB}$$\overrightarrow{AC} = \overrightarrow{OC} - \overrightarrow{OA} = \overrightarrow{AO} + \overrightarrow{OC}$Adding these two equations:$\overrightarrow{AB} + \overrightarrow{AC} = (\overrightarrow{AO} + \overrightarrow{OB}) + (\overrightarrow{AO} + \overrightarrow{OC})$$\overrightarrow{AB} + \overrightarrow{AC} = 2\overrightarrow{AO} + (\overrightarrow{OB} + \overrightarrow{OC})$From the centroid property $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{0}$,we have:$\overrightarrow{OB} + \overrightarrow{OC} = -\overrightarrow{OA} = \overrightarrow{AO}$Substituting this into the sum:$\overrightarrow{AB} + \overrightarrow{AC} = 2\overrightarrow{AO} + \overrightarrow{AO} = 3\overrightarrow{AO}$Comparing this with $\overrightarrow{AB} + \overrightarrow{AC} = n\overrightarrow{AO}$,we get $n = 3$.

$ABC$ is an equilateral triangle. The length of each side is $a$ and the centroid is point $O$. If $\overrightarrow{AB} + \overrightarrow{AC} = n \overrightarrow{AO}$,then find the value of $n$.

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