If $r_1, r_2$ and $r_3$ of a $\triangle A B C$ are in Harmonic progression, then $a, b$ and $c$ will be in

If $r_1, r_2$ and $r_3$ of a $\triangle A B C$ are in Harmonic progression, then $a, b$ and $c$ will be in
  1. arithmetic progression
  2. geometric progression
  3. harmonic progression
  4. arithmetico-geometric progression

Solution

$\because r_1, r_2, r_3$ are in HP. $\begin{array}{ll}\Rightarrow & \frac{2}{r_2}=\frac{1}{r_1}+\frac{1}{r_3} \\ \Rightarrow & \frac{2(s-b)}{\Delta}=\frac{s-a}{\Delta}+\frac{s-c}{\Delta} \\ \Rightarrow & 2 s-2 b=s-a+s-c \\ \Rightarrow & -2 b=-a-c \\ \Rightarrow & 2 b=a+c \\ \Rightarrow \quad & a, b \text { and } c \text { are in } \mathrm{AP} .\end{array}$

Asked in: AP EAMCET 2022 (08 Jul Shift 2)

Practice more Sequences and Series questions on Aicharya