# Thread: Finding a recursive formula for a sequence

1. ## Finding a recursive formula for a sequence

Hi,

I have a sequence:

0, 3, 8, 15, 24

I believe the general formula for the nth term of the sequence is:

tn = (n^2) -1

I cannot figure out what the recursive formula for the sequence is, I appreciate any help.

Thank you

2. ## Re: Finding a recursive formula for a sequence

$s_{k+1} = (\sqrt{s_k+1}+1)^2 - 1$

3. ## Re: Finding a recursive formula for a sequence

Originally Posted by romsek
$s_{k+1} = (\sqrt{s_k+1}+1)^2 - 1$
Ok, I see how that makes sense, would you mind explaining a bit how you arrived at that answer? There aren't any examples in my text that are as complicated as this one.

Thanks again

4. ## Re: Finding a recursive formula for a sequence

Originally Posted by jpompey
Ok, I see how that makes sense, would you mind explaining a bit how you arrived at that answer? There aren't any examples in my text that are as complicated as this one.

Thanks again
$s_k = k^2 -1,~k=1,2, \dots$

$k = \sqrt{s_k + 1}$

$s_{k+1} = (k+1)^2 - 1 = (\sqrt{s_k + 1}+1)^2 - 1$

5. ## Re: Finding a recursive formula for a sequence

$t_n=n^2-1$
$t_{n-1}=(n-1)^2-1$
$t_n-t_{n-1}=n^2-1-((n-1)^2-1)$
$t_n=t_{n-1}+2n-1$

6. ## Re: Finding a recursive formula for a sequence

Originally Posted by jpompey
Hi,

I have a sequence:

0, 3, 8, 15, 24

I believe the general formula for the nth term of the sequence is:

tn = (n^2) -1

I cannot figure out what the recursive formula for the sequence is, I appreciate any help.

Thank you
Notice that 8- 3= 5, 15- 8= 7, 24- 15= 9, 35- 24= 11, etc. That is, the difference between consecutive terms is odd. $t_n- t_{n-1}= 2n- 1$ so that $t_n= t_{n-1}+ 2n- 1$.

7. ## Re: Finding a recursive formula for a sequence

Thanks very much everyone