If the $p^{t h}, q^{t h}$ and $r^{t h}$ terms of a G.P. are a, b and c, respectively. Prove that $a^{q - r} b^{r - p} c^{p - q} = 1$ .

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Answer 1

$L e t f i r s t t e r m A a n d c o m m o n d i f f e r e n c e (R) t_{p} = a t_{q} = b t_{r} = c ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ G . P . t_{p} = A R^{p - 1} = a t_{q} = A R^{q - 1} = b t_{r} = A R^{r - 1} = c t h e n a^{q - r} . b^{r - p} . c^{p - q} {A R^{p - 1}}^{q - r} {A R^{q - 1}}^{r - p} {A R^{r - 1}}^{p - q} A^{(q - r + r - p + p - a)} . R^{{(p - 1) (q - 1) + (q - 1) (r - p) + (r - 1) (p - q)}} A^{\circ} . R^{\circ} = 1 ∴ a^{q - r} . b^{r - p} . c^{p - q} = 1 P r o v e d .$

Answer 2

L e t f i r s t t e r m A a n d c o m m o n d i f f e r e n c e (R) t_{p} = a t_{q} = b t_{r} = c ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ G . P . t_{p} = A R^{p - 1} = a t_{q} = A R^{q - 1} = b t_{r} = A R^{r - 1} = c t h e n a^{q - r} . b^{r - p} . c^{p - q} {A R^{p - 1}}^{q - r} {A R^{q - 1}}^{r - p} {A R^{r - 1}}^{p - q} A^{(q - r + r - p + p - a)} . R^{{(p - 1) (q - 1) + (q - 1) (r - p) + (r - 1) (p - q)}} A^{\circ} . R^{\circ} = 1 ∴ a^{q - r} . b^{r - p} . c^{p - q} = 1 P r o v e d .