# Thread: Confusing Sequence with Absolute value

1. ## Confusing Sequence with Absolute value

Hi Forum!

I'm having difficulties with absolute value numbers.

This sequence is kind of strange, take a look:

Let a be a Real number such that 1<a<2, the sequence {an} is defined by

a1=a and an+1= |an|-1 such that {n=1,2,3,4,...}
And granted that Sn=a1+a2+a3+...+an.

--Find a4, a5, a6 and a7

Ok, no problem with this one
a2=|a1|-1 a2+1=|a1| a1=-a2-1 a2= -a+1
a3=|a2|-1 a3+1=|-a+1| -a3-1=-a+1 a3= a-2
a4=|a3|-1 a4+1=|a-2| -a4-1=a-2 a4= -a+1
...
So, there's an alternate sequence between odd numbered and even numbered terms.
a4=-a+1 a5=a-2
a6=-a+1 a6=a-2

For the problematic part:
Find S2,S4 and S6

S2=a1+a2
S2=a+(-a+1) = 1

So, what's wrong Forum?
I really don't get what is wrong on this one.

If someone has some tips for the absolute value part, I would humbly accept it.
Thanks

2. $\displaystyle a_2=|a_1|-1=|a|-1 =a-1$

This follows because a is between 1 < a < 2

$\displaystyle a_1+a_2=a+(a-1)=2a-1$

3. Hi TheEmptySet!

Things like 1 < a < 2 still confuse me a bit, even more when combined with absolute values.
So, if 1 < a < 2, how come $\displaystyle a_4=-a+1$ ?

I'm learning!!! I'm learning!!!
hahaha I'm loving it.

But still, this is kind of confusing, don't you think?
Thanks again!

4. Originally Posted by Zellator
Hi TheEmptySet!

Things like 1 < a < 2 still confuse me a bit, even more when combined with absolute values.
So, if 1 < a < 2, how come $\displaystyle a_4=-a+1$ ?

I'm learning!!! I'm learning!!!
hahaha I'm loving it.

But still, this is kind of confusing, don't you think?
Thanks again!
You just need to write out the terms!

we know that

$\displaystyle a_3=|a_2|-1$

but we know what $\displaystyle a_2$ is this gives

$\displaystyle a_3=|a-1|-1=(a-1)-1=a-2$

again this follows from 1 < a < 2

but now when you find the next term we get

$\displaystyle a_4=|a_3|-1=|a-2|-1$

but

$\displaystyle a-2 < 0 \implies |a-2|=2-a$

This gives

$\displaystyle a_4=|a_3|-1=|a-2|-1=(2-a)-1=-a+1$

5. Oh, I thought there was a pattern starting in $\displaystyle a_2$.
This relation between absolute values and 1<a<2 are more intimate than I thought.