[tex] S_{10} =100 \\ S_{30} =900. \\ \\ 100 = \frac{ (a_{1}+ a_{10})*10}{2} \Rightarrow 100=(a_{1}+ a_{10})*5 \Rightarrow a_{1}+ a_{10}=20 \\ 900 = \frac{ (a_{1}+ a_{30})*30}{2} \Rightarrow 900=(a_{1}+ a_{30})*15 \Rightarrow a_{1}+ a_{30}=60 \\ \\ a_{1}+ a_{10}=20|*(-1) \\ a_{1}+ a_{30}=60 \Rightarrow a_{10} - a_{30}=40 \Rightarrow a_{1}+9r - a_{1}- 29r = 40 \Rightarrow \\ \Rightarrow 20r = 40 \Rightarrow r = 2 \\ \\ a_{1}+ a_{10}=20 \Rightarrow a_{1}+ a_{1}+9*2 = 20 [/tex]
[tex]\Rightarrow 2 a_{1} = 2 \Rightarrow a_{1} = 1 \\ S_{50}= \frac{ (a_{1}+ a_{50})50 }{2} = \frac{ (a_{1}+ a_{1}+49r)50 }{2}=\frac{ (2+ 98)*50 }{2} = 100*25 = 2500[/tex]
=> [tex] S_{50} = 2500 [/tex]
