CAT 2025 Slot 3 QA Question 6: 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

Question

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

Accepted Answer

Here, $a_n = \left\lfloor\sqrt{n}ightfloor$.
For values of $n$ from 1 to 50:
$a_n = 1$ for $n = 1$ to $3$, contributing $1	imes 3$
$a_n = 2$ for $n = 4$ to $8$, contributing $2	imes 5$
$a_n = 3$ for $n = 9$ to $15$, contributing $3	imes 7$
$a_n = 4$ for $n = 16$ to $24$, contributing $4	imes

For any natural number nnn, let ana_nan​ be the largest integer not exceeding n\sqrt{n}n​. Then the value of a1+a2+⋯+a50a_1 + a_2 + \cdots + a_{50}a1​+a2​+⋯+a50​ is

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