If $p^{t h}, q^{t h}, r^{t h}$ and $s^{t h}$ , terms of an A.P are in G.P, then the show that $(p - q), (q - r), (r - s)$ are also in G.P ?

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Solution

$∵ a_{p} = a + (p - 1) d; a_{q} = a + (q - 1) d; a_{r} = a + (r - 1) d; a_{s} = a + (s - 1) d; A / Q p^{t h}, q^{t h}, r^{t h}, s^{t h} t e r m a r e i n G . P ∴ \frac{a_{q}}{a_{p}} = \frac{a_{r}}{a_{q}} = \frac{a_{s}}{a_{r}} f o r \frac{a_{q}}{a_{p}} = \frac{a_{r}}{a_{q}} \Rightarrow \frac{a_{q}}{a_{p}} - 1 = \frac{a_{r}}{a_{q}} - 1 \Rightarrow \frac{a_{q}}{a_{p}} = \frac{a_{r} - a_{q}}{a_{q} - a_{p}} p u t t i n g t h e v a l u e o f t e r m w e g e t \Rightarrow \frac{a_{q}}{a_{p}} = \frac{(a + (r - 1) d) - (a + (q - 1) d)}{(a + (q - 1) d) - (a + (p - 1) d)} = \frac{q - r}{p - q} ⟶ (1) s i m i l a r l y f o r \frac{a_{r}}{a_{q}} = \frac{a_{s}}{a_{r}} \Rightarrow \frac{a_{r}}{a_{q}} = \frac{r - s}{q - r} ⟶ (2) f r o m (1) a n d (2) w e c o m e t o k n o w (p - q), (q - r) a n d (r - s) a r e i n G . P$

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