Since $a, b, c, d$ are in G.P.
$\begin{aligned}
&\therefore \frac{b}{a}=\frac{c}{b}=\frac{d}{c}=r \Rightarrow b=a r \\
&\begin{array}{l}
c=b r=(a r) r=a r^2 \\
d=c r=\left(a r^2\right) r=a r^3
\end{array} \\
&\text { Now, }\left(a^n+b^n\right),\left(b^n+c^n\right),\left(c^n+d^n\right) \text { are in G.P. } \\
&\text { If }\left(b^n+c^n\right)^2=\left(a^n+b^n\right)\left(c^n+d^n\right) \text { then } \\
&\text { LHS }=\left(b^n+c^n\right)^2=\left[(a r)^n+\left(a r^2\right)^n\right]^2 \\
&\quad=\left[a^n r^n\left(1+r^n\right)\right]^2=a^{2 n} r^{2 n}\left(1+r^n\right)^2 \\
&\text { RHS }=\left(a^n+b^n\right)\left(c^n+d^n\right) \\
&\quad=\left(a^n+a^n r^n\right)\left(a^n r^{2 n}+a^n r^{3 n}\right) \\
&\quad=a^n\left(1+r^n\right) \cdot a^n r^{2 n}\left(1+r^n\right) \\
&\quad=a^{2 n} \cdot r^{2 n}\left(1+r^n\right)^2 \\
&\text { LHS }=\text { RHS. }
\end{aligned}$