# Thread: Sum of a Sequence

1. ## Sum of a Sequence

Hi, so I have a sequence with a rule in recursive form, and I need to find a simple expression for the sequence that is not recursive, which I am having a hard time doing, even though it feels it shouldn't be giving me as much trouble as it is .

Find a simple (non-recursive) expression for the sequence;
$\displaystyle a_{1}$ and $\displaystyle a_{n}=2a_{n-1}+1$ for $\displaystyle n=2,3,4, ....$.

I can't seem to find a pattern here that I could write non-recursively, the terms I found are

$\displaystyle a_{1}=1$ ,
$\displaystyle a_{2}=3$ ,
$\displaystyle a_{3}=7$ ,
$\displaystyle a_{4}=15$ ,
$\displaystyle a_{5}=31$ ,
$\displaystyle a_{6}=63$, ....

Anyone got any tips about what to look for in this sequence? I'm not sure what I'm missing to not see this one non-recusively. Thanks!

2. Originally Posted by superduper
Hi, so I have a sequence with a rule in recursive form, and I need to find a simple expression for the sequence that is not recursive, which I am having a hard time doing, even though it feels it shouldn't be giving me as much trouble as it is .

Find a simple (non-recursive) expression for the sequence;
$\displaystyle a_{1}$ and $\displaystyle a_{n}=2a_{n-1}+1$ for $\displaystyle n=2,3,4, ....$.

I can't seem to find a pattern here that I could write non-recursively, the terms I found are

$\displaystyle a_{1}=1$ ,
$\displaystyle a_{2}=3$ ,
$\displaystyle a_{3}=7$ ,
$\displaystyle a_{4}=15$ ,
$\displaystyle a_{5}=31$ ,
$\displaystyle a_{6}=63$, ....

Anyone got any tips about what to look for in this sequence? I'm not sure what I'm missing to not see this one non-recusively. Thanks!
adding 1 to every term will make things much clearer.

3. Originally Posted by skeeter
adding 1 to every term will make things much clearer.
a-HA!

$\displaystyle a_{n}=2^n - 1,\ where\ a_{1}=1$

I'm awful slow today, thanks!