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Question

# 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
12110
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B
441100
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C
100
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D
110
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Solution

## The correct option is C 100Given k.109=109+2(11)1(10)8+3(11)2(10)7+……+10(11)9 ⇒k=1+2(1110)+3(1110)2+……+10(1110)9……(i) (1110)k=1(1110)+2(1110)9+10(1110)10……(ii) On subtracting Eq. (ii) from Eq. (i), we get k(1−1110)=1+1110+(1110)2+……+(1110)9−10(1110)10 ⇒k(10−1110)=1[(1110)10−1](1110−1)−10(1110)10 [∵In GP, sum of n terms=a(rn−1)r−1, when r>1] ⇒−k=10[10(1110)10−10−10(1110)10] ∴k=100

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