Question

If a, b, c are respectively the $p^{th},q^{th}$ and $r^{th}$ terms of the given G.P., then show that $\log a\left(q-r\right)+\log b\left(r-p\right)+\log c\left(p-q\right)=0$, where $a,b,c>0$.

Answer 1

Let $m$ be the first term and $n$ be the common difference.

$a=m\cdot n^{p-1}$ (1)

$b=m\cdot n^{q-1}$ (2)

$c=m\cdot n^{r-1}$ (3)

Divide equation (2) by (3), (3) by (1) and (1) by (2),

$n^{\left(q-r\right)}=\frac{b}{c}$ (4)

$n^{\left(r-p\right)}=\frac{c}{a}$ (5)

$n^{\left(p-q\right)}=\frac{a}{b}$ (6)

Take log on both sides of equation (4), (5) and (6),

$\left(q-r\right)=\frac{\left(\log b-\log c\right)}{\log n}$ (7)

$\left(r-p\right)=\frac{\left(\log c-\log a\right)}{\log n}$ (8)

$\left(p-q\right)=\frac{\left(\log a-\log b\right)}{\log n}$ (9)

Multiply equation (7) by $\log a$, equation (8) by $\log b$ and equation (9) by $\log c$ and add.

$\log a\left(q-r\right)+\log b\left(r-p\right)+\log c\left(p-q\right)=0$