[tex]\displaystyle a_1=5,~a_{10}=41,~a_6=? \\ a_{10}=41 \Rightarrow a_{10-1}+r=41 \Rightarrow a_9+r=41 \Rightarrow a_1+9r=41 \Rightarrow \\ \Rightarrow 5+9r=41 \Rightarrow 9r=41-5 \Rightarrow 9r=36 \Rightarrow r= \frac{36}{9} \Rightarrow r=4 \\ a_6=a_{6-1}+r \Rightarrow a_6=a_5+r \Rightarrow a_6=a_1+5r \Rightarrow a_6=5+5 \cdot 4 \Rightarrow \\ \Rightarrow a_6=5+20 \Rightarrow \boxed{a_6=25} [/tex]
