Good-Luck Nicolase!
Hrm doing the same thing for an=1+(-1)^N I get
an=1+(-1)^n
a1=1+(-1)^1=0
a2=1+(-1)^2=2
a3=1+(-1)^3=0
a4=1+(-1)^4=2
but when I tried following what you have done I end up, again, unbalanced.
1+(-1)^n=1+(-1)^n+1 (I assumed) Will check in the morning, brain about to explode. thanks again.
This brings up one of the questions that confuses me. Can I use induction to prove recursive? Or are the two completely unrelated? Also. it seems that while I was able to do the work, my remaining confusion seems to be the manner in which the formula is expressed when finished, which seems different than what I started with. I don't believe it matters as the other examples in the section seem to be looking for the ratio: which I believe to be correct. However, I want to know for the sake of knowing because while passing the course is the priority, full understanding is the goal. I dislike being on the outside looking in.
Itried to use induction but it doesn't seem to add up correctly so either I can't do it, or I'm doing something wrong almost immediately.
p(1) 1+2+3...1+(-1)^1=1
0=1 False but true if p(0)
I think it's depends when or where use induction...
Usually in this type of questions better use these steps...
If you had some$\displaystyle a_n=f(n)$ : ($\displaystyle a_n$ is not in recursive form) and you need to find a recursive formula for $\displaystyle a_n$.
So, first of all: Try to find a pattern
second: try to guess recursive formula!
third: Check if your guess been successful guess by "putting" your $\displaystyle a_n=f(n)$
------------------------------------------------------------------------
In your last problem:
$\displaystyle a_n=1+(-1)^n$
so, our sequence is: $\displaystyle 0,2,0,2,0,...$
Finding pattern:
We can see that:
$\displaystyle a_2-a_1=2$
$\displaystyle a_4-a_3=2$
Now is the time for guessing!
We guess that our recursive formula(s) looks like: $\displaystyle a_{n+2}=a_n$
with $\displaystyle a_0=2$ and $\displaystyle a_{n+3}=a_{n+1}$
with $\displaystyle a_0=0$
To prove these formulas you need to substitute $\displaystyle a_n=1+(-1)^n$ in each of them.
I hope you understood something from all this...