Answer
$(6,7,9,11,14,18)$
Work Step by Step
To add two vectors, add each component. This leaves $$\overrightarrow{G}=\overrightarrow{N}+\overrightarrow{M}=(5,6,7,8,9,10)+(1,1,2,3,5,8)$$ $$=(5+1,6+1,7+2,8+3,9+5,10+8)$$ $$=(6,7,9,11,14,18)$$
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