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2011 AMC 10B Problem 10

Problem 10 of 25EasierAlgebraArithmetic

Consider the set of numbers {1,10,102,103,…,1010}.\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?

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Solution

The largest number is 1010.10^{10}. The other ten numbers have sum S=∑i=0910i=1010−19.S=\sum_{i=0}^9 10^i=\dfrac{10^{10}-1}{9}. Therefore their ratio is 1010S=9⋅10101010−1,\dfrac{10^{10}}{S}=9\cdot\dfrac{10^{10}}{10^{10}-1}, which is just slightly greater than 99 and hence closest to 9.9. Thus, the correct answer is B .
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Tagged: geometric sequence · estimation

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