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$If 10^{9} + 2 (11)^{1} (10^{8}) + 3 (11)^{2} (10)^{7} + . . + 10 (11)^{9} = p \cdot 10^{9} then p is$

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a

100

b

110

c

12110

d

441100

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detailed solution

Correct option is A

x=109+2(11)(10)8+…+10(11)9−−−1 Multiplied both sides by 11101110x=11.108+2(11)2(10)7+..+1110−−−21−2⇒x1−1110=109+11(10)8+..+119−1110 =109111010-11110-1-1110 sum of n terms in G.P⇒−x10=1110−1010−1110=−1010x=1011=109×100=p×109⇒p=100

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The positive integer n for which $2 \times 2^{2} + 3 \times 2^{3} + 4 \times 2^{4} + \dots + n \times 2^{n} = 2^{n + 10}$ is

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