Question

If $a=\underset{55\text{times}}{\underbrace{111\dots ..1}}$ , $b=1+10+10^2+10^3+10^4$ and $c=1+10^5+10^{10}+\dots .+10^{50}$ , then

• A. $b,\frac{a}{2},c$ are in arithmetic progression
• B. $b,\sqrt{a},c$ are in geometric progression
• C. $a,b,c$ are in geometric progression
• D. $a,\sqrt{b},c$ are in arithmetic progression

Answer 1

Answer: (B)

$a=1+10+10^2+\dots \dots ..+10^{54}=\frac{10^{55}-1}{9}$
$b=\frac{10^5-1}{9}$
$\begin{array}{rl}c& =\frac{{\left(10^5\right)}^{11}-1}{10^5-1}=\frac{10^{55}-1}{10^5-1}\\ & \Rightarrow \frac{a}{b}=c\\ & \Rightarrow a=bc\\ & \Rightarrow b,\sqrt{a},c\text{ are in G.P. }\end{array}$