Multiple choice

Consider the sequence $2,3,5,6,7,8,10,11,12,13,14,15,17...$ of all positive integers that are not perfect squares. Determine the $2011^{th}$ term of this sequence.

  1. $2056$
  2. $2011$
  3. $2053$
  4. $2055$
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A Correct answer
Explanation

The sequence consists of positive integers excluding perfect squares. To find the 2011th term, we find n such that n - floor(sqrt(n)) = 2011. For n=2056, sqrt(2056) is approx 45.3, so 2056 - 45 = 2011.

AI explanation

If we list all positive integers, the ones missing from the given sequence are exactly the perfect squares. We must find the largest integer k such that k^2 is less than or equal to 2011 plus the number of perfect squares skipped. Taking the square root of 2056 gives roughly 45.3, meaning there are 45 perfect squares before it. Adding 45 skipped squares to the 2011th position gives 2056, confirming it is the 2011th term of the sequence.

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