$\text{ Let }{\mathrm{S}}_1,{\mathrm{S}}_2,{\mathrm{S}}_3\dots \text{ and }{\mathrm{t}}_1,{\mathrm{t}}_2,{\mathrm{t}}_3\dots \text{ are two Arithmatic Sequence such that }$${\mathrm{S}}_1={\mathrm{t}}_1\ne 0,{\mathrm{S}}_2=2{\mathrm{t}}_2\text{ and }\sum _{i=1}^{10} {\mathrm{S}}_{\mathrm{i}}=\sum _{\mathrm{i}=1}^{15} {\mathrm{t}}_{\mathrm{i}}\text{ then }\frac{{\mathrm{S}}_2-{\mathrm{S}}_1}{{\mathrm{t}}_2-{\mathrm{t}}_1}=\frac{{\mathrm{d}}_1}{{\mathrm{d}}_2}\text{ then }{\mathrm{d}}_1-{\mathrm{d}}_2=$

$\text{ Given }{\mathrm{S}}_1+{\mathrm{S}}_2+\dots +{\mathrm{S}}_{10}={\mathrm{t}}_1+{\mathrm{t}}_2+\dots +{\mathrm{t}}_{15}$

$\text{ Let first sequence }{a}_1,a_1+d_1,a_1+2d_1,\dots \text{ and second sequence is }{a}_1,a_1+d_2,a_2+2d_2,\dots {\text{ (sinc s}}_1{\text{=t}}_1\text{)}$

$\begin{array}{l}\text{ Given }{\mathrm{S}}_2=2{\mathrm{t}}_2\\ {\mathrm{a}}_1+{\mathrm{d}}_1=2\left({\mathrm{a}}_1+{\mathrm{d}}_2\right)\\ {\mathrm{a}}_1={\mathrm{d}}_1-2{\mathrm{d}}_2---1\\ \text{ We have to find }\\ \Rightarrow \frac{{\mathrm{S}}_2-{\mathrm{S}}_1}{{\mathrm{t}}_2-{\mathrm{t}}_1}=\frac{{\mathrm{d}}_1}{{\mathrm{d}}_2}\\ \text{ Now }\frac{10}{2}\left[2{\mathrm{a}}_1+9{\mathrm{d}}_1\right]=\frac{15}{2}\left[2{\mathrm{at}}_1+14{\mathrm{d}}_2\right]\\ \Rightarrow {\mathrm{a}}_1=9{\mathrm{d}}_1-21{\mathrm{d}}_2-\cdots -2\\ \text{ For }1\text{ and }2\Rightarrow {\mathrm{d}}_1-2{\mathrm{d}}_2=9{\mathrm{d}}_1-21{\mathrm{d}}_2\\ 8{\mathrm{d}}_1=19{\mathrm{d}}_2\end{array}$

$\frac{{\text{d}}_{\text{1}}}{{\text{d}}_{\text{2}}}=\frac{19}{8}$