If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p - q, q - r, r - s are in G.P.

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Solution

Here, Let R be common ratio,

$a_{p}, a_{q}, a_{r}, a_{s}$ of AP are in GP

$R = \frac{a_{q}}{a_{p}} = \frac{a_{r}}{a_{q}}$

$= \frac{a_{q} - a_{r}}{a_{p} - a_{q}}$ (Ratio property)

$= \frac{[a + (a - 1) d] - [a + (p - 1) d]}{[a + (r - 1) d] - [a + (q - 1) d]}$

$= \frac{(q - r) d}{(p - q) d}$

$R = \frac{q - r}{p - q}$ .......... (i)

Now,

$R = \frac{a_{r}}{a_{q}} = \frac{a_{s}}{a_{r}}$

$= \frac{a_{r} - a_{s}}{a_{q} - a_{r}}$ (Ratio property)

$= \frac{[a + (r - 1) d] - [a + (s - 1) d]}{[a + (q - 1) d] - [a + (r - 1) d]}$

$= \frac{(r - s) d}{(q - r) d}$

$(q - r) d$

$R = \frac{r - s}{q - r}$ ......... (ii)

From equation as (i) and (ii)

$\frac{q - r}{p - q} = \frac{r - s}{q - r}$

$\Rightarrow (p - q), (q - r), (r - s)$ are in GP

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