[tex]\displaystyle a_3=12,~ a_6=30,~a_{10}=? \\ a_3=12 \Rightarrow a_{3-1}+r=12 \Rightarrow a_2+r=12 \Rightarrow a_1+2r=12 \Rightarrow \\ \Rightarrow a_1=12-2r \\ a_6=30 \Rightarrow a_{6-1}+r=30 \Rightarrow a_5+r=30 \Rightarrow a_1+5r=30 \Rightarrow \\ \Rightarrow 12-2r+5r=30 \Rightarrow -2r+5r=30-12 \Rightarrow 3r=18 \Rightarrow \\ \Rightarrow r= \frac{18}{3} \Rightarrow r=6 \\ a_1=12-2r \Rightarrow a_1=12-2 \cdot 12 \Rightarrow a_1=12-12 \Rightarrow a_1=0 [/tex]
[tex]\displaystyle a_{10}=a_{10-1}+r \Rightarrow a_{10}=a_9+r \Rightarrow a_{10}=a_1+9r \Rightarrow a_{10}=0+9 \cdot 6 \Rightarrow \\ \Rightarrow a_{10}=0+54 \Rightarrow \boxed{a_{10}=54}[/tex]
