cat 2024 Question 18

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}+.....+a_{50} is

Accepted Answer

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.

Question 18.