Question

If in an AP, $t_1={\log }_{10}a$, $t_{n+1}={\log }_{10}b$ and $t_{2n+1}={\log }_{10}c$ then $a,b,c$ are in

• A. AP
• B. GP
• C. HP
• D. none of these

Answer 1

The correct option is B GP

Given
$a_1={\log }_{10}a$
$a_1+nd={\log }_{10}b$
$a_1+2nd={\log }_{10}c$
$\Rightarrow nd={\log }_{10}\frac{b}{a}$ and $2nd={\log }_{10}\frac{c}{a}$
$\Rightarrow 2\left({\log }_{10}\frac{b}{a}\right)={\log }_{10}\frac{c}{a}$
$\Rightarrow {\log }_{10}(\frac{b}{a}{)}^2={\log }_{10}\frac{c}{a}$
$\Rightarrow (\frac{b}{a}{)}^2=\frac{c}{a}$
$\Rightarrow b^2=ac$
$\therefore a,b,c$ are in GP