Question

Each of the two triplets of numbers $\log a,\log b,\log c$ and $\log a-\log 2b,\log 2b-\log 3c,\log 3c-\log$ a is an A.P. Check whether the numbers $a,b,c$ be the lengths of the sides of a triangle ?

Answer 1

$\log a,\log b,\log c,\to AP\Rightarrow 2\log b=\log a+\log c\Rightarrow b^2=ac\log a-\log 2b,\log 2b-\log 3c,\log 3c-\log a\to A.P\Rightarrow 2\left(\log 2b-\log 3c\right)=\log a-\log 2b-\log 3c+\log a\Rightarrow 2\log 6bc=\log {6a}^{2}bc\Rightarrow {\left(6bc\right)}^2=\left({6a}^{2}bc\right)\Rightarrow {36b}^2{c}^2={6a}^{2}bc\Rightarrow 6bc=a^2$