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If (10)^9 + 2(11)^1 (10)^8 + 3(11)^2 (1...

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

A

`(121)/(10)`

B

`(441)/(100)`

C

100

D

110

Text Solution

Verified by Experts

The correct Answer is:
C
Given,
`k.10^(9) = 10^(9) + 2(11)^(1) (10)^(8) + 3(11)^(2) (10)^(7) + ..+ 10(11)^(9)`
`rArr k = 1 + 2((11)/(10)) + 3((11)/(10))^(2) + ...+ 10 ((11)/(10))^(9)`...(i)
`((11)/(10))k = 1 ((11)/(10)) + 2 ((11)/(10))^(2) + ...+ 9 ((11)/(10))^(9) + 10 ((11)/(10))^(10)..(ii)
On subtracting Eq. (ii) from Eq. (i), we get
k (1 - (11)/(10)) = 1 + (11)/(10) + ((11)/(10))^(2) + ...+ ((11)/(10))^(9) - 10 ((11)/(10))^(10)`
`rArr k ((10 - 11)/(10)) = (1[((11)/(10))^(10) - 1])/(((11)/(10) -1)) - 10 ((11)/(10))^(10) " " [ :' " In GP, sum of n terms " = (a(r^(n) -1))/(r -1), " when " r gt 1]`
`rArr - k = 10 [10 ((11)/(10))^(10) - 10 - 10 ((11)/(10))^(10)]`
`:. k = 100`
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