If `(10)^(9)+2(11)^(1)(10)^(8)+3(11)^(2)(10)^(7)+"........"+(10)(11)^(9)=k(10)^(9)`, then k is equal to

Options

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

Answer

A

Explanation

The given series can be written as
`k = 1+2(11/10)+3(11/10)^2""+...+9(11/10)^8""+10(11/10)^9" ...(i)"`
On multiplying both sides by `(11/10)`, then
`(11k)/10=(11/10)+2(11/10)^2 +3(11/10)^3""+...+9(11/10)^9""+10(11/10)^10" ...(ii)"`
Now, on subtracting Eq. (ii) frm Eq. (i), then
`-k/10=(11/10)+(11/10)^2""+...+(11/10)^2-10(11/10)^10`
`=(1*{(11/10)^10-1})/((11/10-1))-10(11/10)^10`
`rArr" "k = -100*{(11/10)^10-1}+100(11/10)^10 = 100`