Waste less time on Facebook — follow Brilliant.
×

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
1 year, 7 months ago

No vote yet
1 vote

Comments

Sort by:

Top Newest

Hint: Consider what happens when \( k^2 \leq 2n < (k+1)^2 \). Calvin Lin Staff · 1 year, 7 months ago

Log in to reply

@Calvin Lin 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. Satyajit Mohanty · 1 year, 7 months ago

Log in to reply

@Satyajit Mohanty 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

\[{\left \lceil \dfrac{1 + \sqrt{1+8n}}{2} \right \rceil = \left \lfloor \sqrt{2n} + \frac{3}{2} \right \rfloor}\] Calvin Lin Staff · 1 year, 7 months ago

Log in to reply

if n = 2 it is not true

the only solutions are: 0,1,3,4,6 Fatum Altum · 1 year, 7 months ago

Log in to reply

@Fatum Altum It is true for \(n=2\). I don't agree! Satyajit Mohanty · 1 year, 7 months ago

Log in to reply

@Satyajit Mohanty [] means floor? Fatum Altum · 1 year, 7 months ago

Log in to reply

@Fatum Altum Look. \(\lceil . \rceil\) means ceiling and \(\lfloor . \rfloor\) means floor. Satyajit Mohanty · 1 year, 7 months ago

Log in to reply

@Satyajit Mohanty It's not true when n = 12 (calculated with Wolfram Alpha) Fatum Altum · 1 year, 7 months ago

Log in to reply

@Satyajit Mohanty Ah- i see =) Fatum Altum · 1 year, 7 months ago

Log in to reply

@Fatum Altum It's not true when n = 12 (calculated with Wolfram Alpha) Fatum Altum · 1 year, 7 months ago

Log in to reply

Comment deleted Jul 29, 2015

Log in to reply

@Satyajit Mohanty 5 != 4 Fatum Altum · 1 year, 7 months ago

Log in to reply

@Fatum Altum Can you give me the set of values of \(n\) for which the equation doesn't satisfy? Satyajit Mohanty · 1 year, 7 months ago

Log in to reply

@Satyajit Mohanty 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 Fatum Altum · 1 year, 7 months ago

Log in to reply

@Satyajit Mohanty Sure, wait a minute Fatum Altum · 1 year, 7 months ago

Log in to reply

@Fatum Altum I'm sorry. I agree. Satyajit Mohanty · 1 year, 7 months ago

Log in to reply

@Satyajit Mohanty Can you look at my problem, please ? Fatum Altum · 1 year, 7 months ago

Log in to reply

False. Often, they don't agree. Michael Mendrin · 1 year, 7 months ago

Log in to reply

@Michael Mendrin I'm sorry. I forgot to mention that \(n\) is a positive integer! Can you find some counter-example now? Satyajit Mohanty · 1 year, 7 months ago

Log in to reply

@Satyajit Mohanty ..

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

But because of the way ceiling and floor functions work, these two go off in opposite directions. This is just one example. Michael Mendrin · 1 year, 7 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...