If `(10)^9+2(11)^2(10)^7 +….+10 (11)^9 = k(10)^9`

Options

A. 100
B. 110
C. `(121)/(10)`
D. `(441)/(100)`

Answer

A

Explanation

Let
`S=(10)^(9)+2(11)^(1)(10)^(8)+3(11)^(2)(10)^(7)+ . . . .+10(11)^(9)`. Then,
`(11)/(10)S=11(10)^(8)+2(11)^(2)(10)^(7)+ . . . .+9(11)^(9)+(11)^(10)`
`:." "S-(11)/(10)S=10^(9)+{(11)(10)^(8)+(11)^(2)(10)^(7)+ . . .+(11)^(9)}-(11)^(10)`
`rArr-(S)/(10)=(10^(9){1-((11)/(10))^(10)})/(1-(11)/(10))-(11)^(10)`
`rArr-(S)/(10)=-10^(10){1-((11)/(10))^(10)}-(11)^(10)`
`rArr" "S=10^(11)-10(11)^(10)+10(11)^(10)=10^(11)=100(10^(9))`
`rArr" "k(10^(9))=100(10^(9))rArrk=100`