Sequence and Series

Concept:If four terms of an AP are in GP, then the differences of their indices are also in GP.Explanation:Let the AP have first term $a$ and common difference $d$ ($d \neq 0$).Then the $p^\text{th}$, $q^\text{th}$, $r^\text{th}$, $s^\text{th}$ terms are:$T_p = a + (p-1)d$, $T_q = a + (q-1)d$, $T_r = a + (r-1)d$, $T_s = a + (s-1)d$.Given these are in GP, so $\frac{T_q}{T_p} = \frac{T_r}{T_q} = \frac{T_s}{T_r} = R$ (common ratio).Let $a + (p-1)d = A$, then $T_q = AR$, $T_r = AR^2$, $T_s = AR^3$.Subtracting consecutive terms gives:$T_q - T_p = A(R-1)$$T_r - T_q = AR(R-1)$$T_s - T_r = AR^2(R-1)$But $T_q - T_p = (q-p)d$, $T_r - T_q = (r-q)d$, $T_s - T_r = (s-r)d$.So $(q-p)$, $(r-q)$, $(s-r)$ are proportional to $1$, $R$, $R^2$, hence they are in GP.Thus $p-q = -(q-p)$, $q-r = -(r-q)$, $r-s = -(s-r)$ are also in GP (negation preserves the common ratio).Answer:G.P.