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

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Question

(a) If $a$ and $d$ be the first term and common difference of the $A.P.$

Then ${T_p} = a + (p - 1)d,\;$

${T_q} = a + (q - 1)d$

and ${T_r} = a

Vedclass · Accepted Answer

$G.P.$

Vedclass · Answer

(a) If $a$ and $d$ be the first term and common difference of the $A.P.$

Then ${T_p} = a + (p - 1)d,\;$

${T_q} = a + (q - 1)d$

and ${T_r} = a + (r - 1)d$.

If ${T_p},\;{T_q},\;{T_r}$ are in $G.P.$

Then its common ratio $R = \frac{{{T_q}}}{{{T_p}}} = \frac{{{T_r}}}{{{T_q}}} = \frac{{{T_q} - {T_r}}}{{{T_p} - {T_q}}}$

$ = \frac{{[a + (q - 1)d] - [a + (r - 1)d]}}{{[a + (p - 1)d] - [a + (q - 1)d]}} = \frac{{q - r}}{{p - q}}$

Similarly, we can show that $R = \frac{{q - r}}{{p - q}} = \frac{{r - s}}{{q - r}}$

Hence $(p - q),\;(q - r),\;(r - s)$ be in $G.P.$

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

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