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. $\frac{121}{10}$
• B. $\frac{441}{100}$
• C. $100$
• D. $110$

Answer 1

Answer: (C)

$100$

Answer 2

$10^{9}+2 \cdot(11)(10)^{8}+3(11)^{2}(10)^{7}+\ldots+10(11)^{9}=k(10)^{9}$
$x=10^{9}+2 \cdot(11)(10)^{8}+3(11)^{2}(10)^{7}+\ldots+10(11)^{9}$
$\frac{11}{10} x=11 \cdot 10^{8}+2 \cdot(11)^{2} \cdot(10)^{7}+\ldots+9(11)^{9}+11^{10}$

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$x\left(1-\frac{11}{10}\right)=10^{9}+11(10)^{8}+11^{2} \times(10)^{7}+\ldots+11^{9}-11^{10}$
$\Rightarrow -\frac{x}{10}=10^{9}\left(\frac{\left(\frac{11}{10}\right)^{10}-1}{\frac{11}{10}-1}\right)-11^{10}$
$\Rightarrow-\frac{x}{10}=\left(11^{10}-10^{10}\right)-11^{10}=-10^{10}$
$\Rightarrow x=10^{11}=k \cdot 10^{9}$
$\Rightarrow \,k=100$