If the p^{th}, q^{th}, r^{th} and s^{th} term | vedclass

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(A) Let $a$ be the first term and $d$ be the common difference of the $A.P.$The terms are $T_p = a + (p - 1)d, T_q = a + (q - 1)d, T_r = a + (r - 1)d,

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$G.P.$

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(A) Let $a$ be the first term and $d$ be the common difference of the $A.P.$The terms are $T_p = a + (p - 1)d, T_q = a + (q - 1)d, T_r = a + (r - 1)d, T_s = a + (s - 1)d$.Since $T_p, T_q, T_r, T_s$ are in $G.P.$,the common ratio $R$ is given by:$R = \frac{T_q}{T_p} = \frac{T_r}{T_q} = \frac{T_s}{T_r} = \frac{T_q - T_r}{T_p - T_q} = \frac{T_r - T_s}{T_q - T_r}$.Substituting the values:$T_q - T_r = (q - r)d$ and $T_p - T_q = (p - q)d$.Thus,$R = \frac{(q - r)d}{(p - q)d} = \frac{q - r}{p - q}$.Similarly,$R = \frac{r - s}{q - r}$.Since $\frac{q - r}{p - q} = \frac{r - s}{q - r}$,the terms $(p - q), (q - r), (r - s)$ are in $G.P.$

If the $p^{th}, q^{th}, r^{th}$ and $s^{th}$ terms of an $A.P.$ are in $G.P.$,then $(p - q), (q - r), (r - s)$ will be in:

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