$$ \begin{aligned} & S_{10}=390 \\ & \frac{10}{2}[2 a+(10-1) d]=390 \\ & \Rightarrow 2 a+9 d=78 \\ & \frac{t_{10}}{t_5}=\frac{15}{7} \Rightarrow \frac{a+9 d}{a+4 d}=\frac{15}{7} \Rightarrow 8 a=3 d \end{aligned} $$ From (1) & (2) $a=3 \& d=8$ $$ \begin{aligned} & S_{15}-S_5=\frac{15}{2}(6+14 \times 8)-\frac{5}{2}(6+4 \times 8) \\ & =\frac{15 \times 118-5 \times 38}{2}=790 \end{aligned} $$