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## Loser66 one year ago Conjecture the formula for the nth term of {a_n} if the first ten terms of it is 1,0,0,1,0,0,0,0,1,0 Please, help

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1. Loser66

I want to use the floor function to put it in neat. Obviously, |dw:1440875326682:dw|

2. Loser66

But how?

3. freckles

what do you mean how if you say it is obvious?

4. freckles

is the piecewise part obvious but not how to write in the floor format? is that what you?

5. freckles

is that what you mean*?

6. ikram002p

oh that remind me of Legendre function

7. Loser66

When I say "obvious" , it means we can see it obviously.

8. freckles

ok then what do you need help on then

9. Loser66

I want the floor format.

10. freckles

oh okay

11. Loser66

$$\lfloor{\sqrt n}\rfloor$$

12. Loser66

How about this? $$1<\sqrt 2 < 2$$, hence $$\lfloor{\sqrt 2}\rfloor =1$$ and $$\lfloor{\dfrac{\lfloor{\sqrt 2}\rfloor}{\sqrt2}}\rfloor=0$$

13. Loser66

It works well for n =3,4, 5, 6... but how can I just list them out like this? I want a general logic to lead me there.

14. ikram002p

so sqrt(n^2) is integer, then [sqrt(n^2) ] would be same integer n else u would have an additional decimals there are two lovely things about floor function 1-[sqrt(n^2) ]<= sqrt(n^2) 2-[r]=0 if 0<r<1 and you already know [sqrt(n^2) ]<= sqrt(n^2) $$\dfrac {[\sqrt(n^2) ]}{\sqrt(n^2)}<=1$$ we have two cases case 1 $$\dfrac {[\sqrt(n^2) ]}{\sqrt(n^2)}=1$$ which mean n is an integer and n^2 is a square of integer also $$[\dfrac {[\sqrt(n^2) ]}{\sqrt(n^2)}] =1$$ case 2 $$\dfrac {[\sqrt(n^2) ]}{\sqrt(n^2)}<1$$ from property 2 $$[\dfrac {[\sqrt(n^2) ]}{\sqrt(n^2)}] =0$$ which u already mean that n is not a square of integer

15. ikram002p

and you can use n =i^2 instead n^2 as the general term lol

16. Loser66

Hey, friend , I don't get why $$\sqrt n^2$$. Is it not that it is always = n and n >0 since it is the order of the term.

17. Loser66

If we consider $$\lfloor{\sqrt n^2}\rfloor = \lfloor{n}\rfloor$$ and it is = n itself.

18. Loser66

since $$n\in \mathbb N$$

19. ikram002p

yes u are correct , its just a variable :D sorry to confuse u ;)

20. ikram002p

i just used it to explain but since u were asked to make general form of n then use n without n^2

21. Loser66

oh, I got you. Thank you so much.

22. ikram002p

you are the most wlc <3

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