# Thread: formula for a sequence

1. ## formula for a sequence

i am looking for an explicit formula for the sequence given by the following numbers:

0, 1/2, 1, 2/3, 1/3, 0, 1/4, 2/4, 3/4, 1, 4/5, 3/5, 2/5, 1/5, 0, 1/6, etc..

2. I could get upto this:

nth term,$\displaystyle t_n=\frac{d-m}{d}$

where
$\displaystyle d=\left\lceil\frac{\sqrt{8n+1}-1}{2}\right\rceil$

$\displaystyle m=(-1)^d \left( \frac {d(d+(-1)^d)}{2}-n \right)$

3. $\displaystyle d_{n}=\left\lfloor\sqrt{2n}+\frac{1}{2}\right\rflo or$

4. we get the general term of the sequence to be:

$\displaystyle a_{n}$=$\displaystyle \sum_{n=1}^{n}\frac{(-1)^{d_{n}}}{d_{n}}\$

i figured this out yesterday but didn't have the time to learn LaTex. And I had Mr. Fantastic sending me demerits all day long. maybe he could have spent more time helping with this as opposed to sitting in front of his computer sending out demerits!!

5. i will probably get thrown off the site now!! well it was fun learning a little bit of latex!

6. oh yes, one more thing,

$\displaystyle a_{0}$=0.

7. Originally Posted by kkoutsothodoros
we get the general term of the sequence to be:

$\displaystyle a_{n}$=$\displaystyle \sum_{n=1}^{n}\frac{(-1)^{d_{n}}}{d_{n}}\$

8. it does seem a little unclear. sorry, i am just learning how to use latex.

let $\displaystyle a_{n}$ = $\displaystyle \frac{(-1)^{d_{n}}}{d_{n}}\$ for n >= 2, $\displaystyle a_{1}$ = 0.

let $\displaystyle s_{n}\$ = $\displaystyle \sum_{n=1}^{n}a_{n}$.

you should get

$\displaystyle s_{1}$ = 0
$\displaystyle s_{2}$ = .5
$\displaystyle s_{3}$ = 1
$\displaystyle s_{4}$ = 2/3
etc.

9. so $\displaystyle s_{n}$ is the general term for the sequence 0, 1/2, 2/2, 2/3, 1/3, 0/3, 1/4, 2/4, 3/4, 4/4, 4/5, 3/5, etc

10. the funny thing is this sequence is just used as a counterexample in a cauchy sequence problem i was looking at. it wasn't that important to get the general term formula. it just bugged me so much that i couldn't let it go. but i guess it's useful that i learned how to get a general term for 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, etc along the way. the general term for this is:

$\displaystyle d_{n}=\left\lfloor\sqrt{2n}+\frac{1}{2}\right\rflo or$

11. Originally Posted by malaygoel
is it more clear now?

12. Originally Posted by kkoutsothodoros
the funny thing is this sequence is just used as a counterexample in a cauchy sequence problem i was looking at. it wasn't that important to get the general term formula. it just bugged me so much that i couldn't let it go. but i guess it's useful that i learned how to get a general term for 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, etc along the way. the general term for this is:

$\displaystyle d_{n}=\left\lfloor\sqrt{2n}+\frac{1}{2}\right\rflo or$
your expression and my expession for d(given in post #2) are quite different. And they both work. Can you explain how you got your expression. mine had a simple logic.

13. Consider 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,etc

where the integer m appears m times in the sequence. using the fact that

$\displaystyle 1+2+3+...+n=\frac{n(n+1)}{2}\$ we have $\displaystyle \frac{m(m-1)}{2}\$ < n <= $\displaystyle \frac{m(m+1)}{2}\$

m < $\displaystyle \sqrt{2n}+\frac{1}{2}\$ <= m + 1 which is equivalent to what we are looking for.

14. Originally Posted by kkoutsothodoros
Consider 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,etc

where the integer m appears m times in the sequence. using the fact that

$\displaystyle 1+2+3+...+n=\frac{n(n+1)}{2}\$ we have $\displaystyle \frac{m(m-1)}{2}\$ < n <= $\displaystyle \frac{m(m+1)}{2}\$
Upto here, everything is correct.

Now, how do you get this

m < $\displaystyle \sqrt{2n}+\frac{1}{2}\$ <= m + 1

15. sorry again, this part is a little tricky and i found an error.

multiply $\displaystyle \frac{m(m-1)}{2}\$ < n <= $\displaystyle \frac{m(m+1)}{2}\$ by 2 to get $\displaystyle m^2 - m < 2n <=m^2 + m$. now add 1/4 to both sides and factor to get $\displaystyle m^2 - m + 1/4< 2n + 1/4 <=m^2 + m + 1/4$ or $\displaystyle (m-1/2)^2 < 2n + 1/4 <= (m + 1/2)^2$. take the square root of everything and we get $\displaystyle (m-1/2) < \sqrt{2n + 1/4}\ <= (m + 1/2)$.

add 1/2 to both sides and we get $\displaystyle m < \sqrt{2n + 1/4}\ + 1/2 <= m + 1$.

i just checked it in Excel and the 1/4 is not affecting the result, luckily. it still works if we start with n = 0.

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