CAT 2024 Slot 1 QA Q18 Number Systems

Question 18 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.

For any natural number \(n\), let \(a_n\) be the largest integer not exceeding \(\sqrt{n}\). Then the value of \(a_1+a_2+\cdots+a_{50}\) is

TITA: type in the answer. There are no options and no negative marking.

Answer: 217 Number Systems TITA

Solution

\(a_n = k\) for \(k^2 \le n \le (k+1)^2 - 1\), which is \(2k + 1\) values of \(n\).
\(k = 1\) to \(6\) cover \(n = 1\) to \(48\): \(\sum k(2k+1) = 3 + 10 + 21 + 36 + 55 + 78 = 203\).
\(n = 49, 50\) give \(a_n = 7\): \(14\) more.
Total \(= 203 + 14 = 217\).