If `a = underset(55 "times")underbrace(111.....1), b= 1+10+10^(2)+10^(3)+10^(4) and c=1+10^(5)+10^(10)+.+10^50` then prove that a=bc

Options

A. `b, (a)/(2), c` are in `A.P`
B. `b, sqrt(a), c` are in `G.P`
C. `a` is `a` prime number
D. `a` is `a` composite number

Answer

B, D

Explanation

If `a = underset(55 "times")underbrace(111....1)` ………….
`a = underset(55 "times")underbrace(111....1) = 1+10+10^(2)+……+10^(54)= (10^(55)-1)/(9)`
`b=1+10+10^(2)+10^(2)+10^(3)+10^(4)= (10^(5)-1)/(9)`
`c=1+10^(5)+10^(10)+……+10^(50)`
`= ((10^(5))^(11)-1)/(10^(5)-1)=(10^(55)-1)/(10^(5)-1)=(((10^(55)-1)/9)/(10^(5)-1))/(9) =(a)/(b)`
`a = bc :. b, sqrt(a) , c are in G.P`