2011 AMC 10B Problem 10

Below is the professionally curated solution for Problem 10 of the 2011 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 10B solutions, or check the answer key.

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Concepts:geometric sequence estimation

Difficulty rating: 1280

10.

Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}.$ 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?

$1$

$9$

$10$

$11$

$101$

Solution:

The largest number is $10^{10}.$ The rest of the number have a sum of $S=\sum_{i=0}^9 10^i.$ Then, $10S = \sum_{i=0}^9 10^{i+1},$ making $9S = 10^{10}-1.$ This means that $\dfrac{10^{10}}{9S}$ is close to one, so the ratio between $10^{10}$ and the sum is close to $9.$

Thus, the correct answer is B .

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