Question

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

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