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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)=10^(9) +2(11)^(1)(10)^(8)+3(11)^(2)(10)^(7)` ,
` +...+10(11)^(9)`
` 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(11/9)^(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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