[tex]\displaystyle a_2=4,~a_5=25,~a_1=?,~r=?,~S_{20}=? \\ \boxed{a_n=a_{n-1}+r} \\ a_2=4 \Rightarrow a_{2-1}+r=4 \Rightarrow a_1+r=4 \Rightarrow a_1=4-r \\ a_5=25 \Rightarrow a_{5-1}+r=25 \Rightarrow a_4+r=25 \Rightarrow a_1+4r=25 \Rightarrow \\ \Rightarrow 4-r+4r=25 \Rightarrow 4+3r=25 \Rightarrow 3r=25-4 \Rightarrow \\ \Rightarrow 3r=21 \Rightarrow r= \frac{21}{3} \Rightarrow \boxed{r=7} \\ a_1=4-r \Rightarrow a_1=4-7 \Rightarrow \boxed{a_1=-3} \\ \boxed{S_n= \frac{2a_1+(n-1) \cdot r}{2} \cdot n} [/tex]
[tex]\displaystyle S_{20}= \frac{2 \cdot (-3)+(20-1) \cdot 7}{\not2} \cdot \not20 \\ S_{20}= (-6+19 \cdot 7) \cdot 10 \\ S_{20}=(-6+133) \cdot 10 \\ S_{20}=127 \cdot 10 \\ \boxed{S_{20}=1270} [/tex]
