# A problem on Floor and Ceiling Functions!

Is it true or false?

$\large{\left \lceil \dfrac{1 + \sqrt{1+8n}}{2} \right \rceil - 1 = \left \lfloor \sqrt{2n} \right \rfloor}$

Here $$n$$ is a positive integer.

If it is false, provide me a counter-example. If it's true, please provide me a proof.

Note by Satyajit Mohanty
2 years, 11 months ago

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Hint: Consider what happens when $$k^2 \leq 2n < (k+1)^2$$.

Staff - 2 years, 11 months ago

Thanks :) I got it. I actually had a doubt on a solution to this problem: Peculiar Sequence of Positive Integers! as the generalized version of my solution did not match the generalized version of the other solution.

- 2 years, 11 months ago

If you see my note, I was pointing out the mistake that @Chew-Seong Cheong made in the generalization.

The true version of the statement that you are looking for, is

${\left \lceil \dfrac{1 + \sqrt{1+8n}}{2} \right \rceil = \left \lfloor \sqrt{2n} + \frac{3}{2} \right \rfloor}$

Staff - 2 years, 11 months ago

if n = 2 it is not true

the only solutions are: 0,1,3,4,6

- 2 years, 11 months ago

It is true for $$n=2$$. I don't agree!

- 2 years, 11 months ago

[] means floor?

- 2 years, 11 months ago

Look. $$\lceil . \rceil$$ means ceiling and $$\lfloor . \rfloor$$ means floor.

- 2 years, 11 months ago

It's not true when n = 12 (calculated with Wolfram Alpha)

- 2 years, 11 months ago

Ah- i see =)

- 2 years, 11 months ago

It's not true when n = 12 (calculated with Wolfram Alpha)

- 2 years, 11 months ago

Comment deleted Jul 29, 2015

5 != 4

- 2 years, 11 months ago

Can you give me the set of values of $$n$$ for which the equation doesn't satisfy?

- 2 years, 11 months ago

So, the first n is 4; then I tried to see the connection, here it is : (11 , 12) then we adding 4 and get the next pair(16,17) - two pairs of n that don't satisfy; then we adding 5 and get triplet (22 , 23 , 24) again 5 and (29,30,31) - two triplets for n; adding 6 and get quartet (37.38,39,40) again 6 and (46,47,48,49) - two quartets; and so on (next 7 - 2 pairs of 5 pieces of n ) to infinity

- 2 years, 11 months ago

Sure, wait a minute

- 2 years, 11 months ago

I'm sorry. I agree.

- 2 years, 11 months ago

Can you look at my problem, please ?

- 2 years, 11 months ago

False. Often, they don't agree.

- 2 years, 11 months ago

I'm sorry. I forgot to mention that $$n$$ is a positive integer! Can you find some counter-example now?

- 2 years, 11 months ago

..

$$\dfrac { 1 }{ 2 } (1+\sqrt { 1+8\cdot 4 } )-1=2.37228...$$
$$\sqrt { 2\cdot 4 } =2.828427...$$

But because of the way ceiling and floor functions work, these two go off in opposite directions. This is just one example.

- 2 years, 11 months ago