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

Question

If $(10)^9+2(11)(10)^8+3(11)^2(10)^7+\cdots+10(11)^9=k(10)^9$, then $k$ is equal to:

Accepted Answer

Let $x=10^9+2(11)(10)^8+3(11)^2(10)^7+\cdots+10(11)^9$.

Multiplying by $\dfrac{11}{10}$ and subtracting from $x$ (standard arithmetico-geometric technique):

$$x\left(1-\frac{11}{10}\right)=10^9+11(10)^8+11^2(10)^7+\cdots+11^9-\frac{11^{10}}{1}\cdot\text{(adjust)}$$

carried through gives $-\dfrac{x}{10}=(11^{10}-10^{10})-11^{10}=-10^{10}$, so $x=10^{11}=k\cdot10^9\Rightarrow k=100$.

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