[tex]\displaystyle a_7=22,~a_{20}=74,~a_{10}=? \\ \\ a_7=22 \Rightarrow a_{7-1}+r=22 \Rightarrow a_6+r=22 \Rightarrow a_1+6r=22 \Rightarrow \\ \\ \Rightarrow a_1=22-6r \\ \\ a_{20}=74 \Rightarrow a_{20-1}+r=74 \Rightarrow a_{19}+r=74 \Rightarrow a_1+19r=74 \Rightarrow \\ \\ \Rightarrow 22-6r+19r=74 \Rightarrow -6r+19r=74-22 \Rightarrow 13r=52 \Rightarrow \\ \\ \Rightarrow r= \frac{52}{13} \Rightarrow r=4 \\ \\ a_1=22-6r \Rightarrow a_1=22-6 \cdot 4 \Rightarrow a_1=22-24 \Rightarrow a_1=-2 [/tex]
[tex]\displaystyle a_{10}=a_{10-1}+r \Rightarrow a_{10}=a_9+r \Rightarrow a_{10}=a_1+9r \Rightarrow \\ \\ \Rightarrow a_{10}=-2+9 \cdot 4 \Rightarrow a_{10}=-2+36 \Rightarrow \boxed{a_{10}=34}[/tex]
