$\frac{{ }^7{\mathrm{C}}_1}{{ }^7{\mathrm{C}}_1{+}^7{\mathrm{C}}_2},\hspace{0.33em}\hspace{0.33em}\frac{{ }^7{\mathrm{C}}_2}{{ }^7{\mathrm{C}}_2{+}^7{\mathrm{C}}_3},\hspace{0.33em}\hspace{0.33em}\frac{{ }^7{\mathrm{C}}_3}{{ }^7{\mathrm{C}}_3{+}^7{\mathrm{C}}_4}$ are in 

$$\begin{gathered} \frac{{ }^7{\mathrm{C}}_1}{{ }^7{\mathrm{C}}_1{+}^7{\mathrm{C}}_2}=\frac{1}{4}\Rightarrow \frac{{ }^7{\mathrm{C}}_1}{{ }^8{\mathrm{C}}_2}=\frac{1}{4}\Rightarrow \frac{{ }^7{\mathrm{C}}_1}{{\displaystyle \frac{8}{2}}\left({ }^7{\mathrm{C}}_1\right)}=\frac{1}{4},\hspace{0.33em}\hspace{0.33em} \\ \frac{{ }^7{\mathrm{C}}_2}{{ }^7{\mathrm{C}}_2{+}^7{\mathrm{C}}_3}=\frac{3}{8},\hspace{0.33em}\hspace{0.33em}\frac{{ }^7{\mathrm{C}}_3}{{ }^7{\mathrm{C}}_3{+}^7{\mathrm{C}}_4}=\frac{1}{2} \\ \frac{1}{4},\frac{3}{8},\frac{1}{2} are in A.P \end{gathered}$$