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

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

A

100

B

110

C

`(121)/(10)`

D

`(441)/(100)`

Text Solution

Verified by Experts

The correct Answer is:
A
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`
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