CAT 2024

Slot 1

QA

Question & Solution

We are told that, for any natural number $n_{1}$ let $a_{n}$ be the largest integer not exceeding $\sqrt{n}$

So for n=1, the largest integer not exceeding $\sqrt{1}$ will be 1
For n=2, the largest integer not exceeding $\sqrt{2}$ will be 1
For n=3, the largest integer not exceeding $\sqrt{3}$ will be 1
For n=4, the largest integer not exceeding $\sqrt{4}$ will be 2

We see a pattern here regarding the squares of the numbers, 
Listing down all the perfect squares, 
1, 4, 9, 16, 25, 36, 49, 64, ...
We see that the difference between 4 and 1 is 3 and there were three natural numbers in the given pattern with the value as 1, 
So we can write for the rest of the numbers as well, 
3 numbers will have value 1, giving a total value of 3
5 numbers will have value 2, giving a total value of 10
7 numbers will have value 3, giving a total value of 21
9 numbers will have value 4, giving a total value of 36
11 numbers will have value 5, giving a total value of 55
13 numbers will have value 6, giving a total value of 78
Now, only the values of $a_{49},\ a_{50}$ will have the value of 7, total value of 14. 

Adding these values, we get the total sum as 217, which is the answer.