1 step: [tex]S_{5}=T_{1}+T_{2}+T_{3}+T_{4}+T_{5}[/tex], [tex]S_{3}=T_{1}+T_{2}+T_{3}[/tex], then
[tex]S_{5}-S_{3}=T_{4}+T_{5}=5[/tex].
2 step: [tex]T_{n}=T_{1}*q^{n-1}[/tex], then
[tex]T_{6}=T_{1}*q^{5}[/tex]
[tex]T_{5}=T_{1}*q^{4}[/tex]
[tex]T_{4}=T_{1}*q^{3}[/tex]
[tex]T_{3}=T_{1}*q^{2}[/tex]
and [tex] \left \{ {{T_{6}-T_{4}= \frac{5}{2} } \atop {T_{5}+T_{4}=5}} \right. [/tex] will have form [tex] \left \{ {{T_1*q^{5}-T_{1}*q^{3}= \frac{5}{2} } \atop {T_{1}*q^{4}+T_{1}*q^{3}=5} \right. [/tex].
3 step: Solve this system [tex] \left \{ {{T_1*q^{3}*(q^{2}-1)= \frac{5}{2} } \atop {T_{1}*q^{3}*(q+1)=5} \right. [/tex] and dividing first equation on second we obtain [tex] \frac{q^{2}-1}{q+1}= \frac{ \frac{5}{2} }{5} [/tex]. So, [tex] \frac{(q-1)(q+1)}{q+1} = \frac{1}{2} [/tex] and [tex]q-1= \frac{1}{2} [/tex], [tex]q= \frac{3}{2} [/tex] - the common ratio.
4 step: Insert [tex]q= \frac{3}{2} [/tex]into equation [tex]T_{1}*q^{3}*(q+1)=5[/tex] and obtain [tex]T_{1}* \frac{27}{8}*( \frac{3}{2}+1 ) =5[/tex], from where [tex]T_{1}= \frac{16}{27} [/tex].
