Thread: Finding the recursive definition for an integer sequence

1. Finding the recursive definition for an integer sequence

I'm not sure I fully understand how to do a recursive definition for an integer sequence containing an exponent.

$a_n = 4^n + 12; \forall n \geq 0$

I can figure out that:

$4^0 + 12 = 13$
$4^1 + 12 = 16$
$4^2 + 12 = 28$
$4^3 + 12 = 76$
$4^4 + 12 = 268$

which equates to (mutiples of 4) x 12...but I'm not sure how to finish this to make it a recursive definition. The question I'm trying to answer has different values, my hope is to learn how to do this in general and not the specific question I'm required to answer.

Thank you.

2. Re: Finding the recursive definition for an integer sequence

You could write:

$a_{n}=4^{n}+12$

$a_{n+1}=4^{n+1}+12$

Now, subtract the former from the latter to get a recursion (difference equation).

3. Re: Finding the recursive definition for an integer sequence

Originally Posted by Kevmck
I'm not sure I fully understand how to do a recursive definition for an integer sequence containing an exponent.
$a_n = 4^n + 12; \forall n \geq 0$
I can figure out that:
$4^0 + 12 = 13$
$4^1 + 12 = 16$
$4^2 + 12 = 28$
$4^3 + 12 = 76$
$4^4 + 12 = 268$
which equates to (mutiples of 4) x 12...but I'm not sure how to finish this to make it a recursive definition. The question I'm trying to answer has different values, my hope is to learn how to do this in general and not the specific question I'm required to answer.
I do not understand what you are asking.
What do you want done here?

4. Re: Finding the recursive definition for an integer sequence

Originally Posted by MarkFL2
You could write:

$a_{n}=4^{n}+12$

$a_{n+1}=4^{n+1}+12$

Now, subtract the former from the latter to get a recursion (difference equation).
Thank you for the quick response.

I'm guessing here, but that would look something like
$4^n^+^1 + 12$
$- 4^n + 12$

And the only answer I seem to see right now is wrong:
$4^n^+^1 - 4^n = 1$
but that doesn't seem like it makes any sense...

5. Re: Finding the recursive definition for an integer sequence

Originally Posted by Plato
I do not understand what you are asking.
What do you want done here?
The question asks me to give the recursive definition for the integer sequence. The only example I have to work off of is:
$a_n = 4n - 1, n \geq 1$
$a_n = 4n - 1$
$a_1_0_0 = 4(100) - 1 = 399$
$a_n_-_1 = 4(n-1) - 1 = 4n - 5$
$a_n_-_1 = a_n - 4$
$a_n = a_n_-_1 + 4; a_1 = 3; n \geq 2$

6. Re: Finding the recursive definition for an integer sequence

What I am suggesting would give you:

$a_{n+1}=a_n+3\cdot4^n$ where $a_0=13$

7. Re: Finding the recursive definition for an integer sequence

I thank you for the explanation, but I must be doing something wrong. When I use that formula I seem to be getting different answers.
$a_1_+_1 = 13 + 3 \cdot 4^1$ where $a_1 = 25$

8. Re: Finding the recursive definition for an integer sequence

$a_{n+1}=4^{n+1}+12$

$a_{n}=4^{n}+12$

Subtracting the bottom from the top, we find:

$a_{n+1}-a_{n}=4^{n+1}-4^{n}+12-12$

$a_{n+1}-a_{n}=4^{n}(4-1)$

$a_{n+1}=a_{n}+3\cdot4^{n}$

Now, from the definition you provided, we find:

$a_{0}=4^0+12=13$

Hence, using the recursion we have found we get:

$a_{1}=16$
$a_{2}=28$
$a_{3}=76$
$a_{4}=268$

9. Re: Finding the recursive definition for an integer sequence

Thank you MarkFL2. I'm starting to understand this now. I appreciate your help.

10. Re: Finding the recursive definition for an integer sequence

Finding an explicit formula for $a_n$ when $a_n$ is defined by recursion is a meaningful problem. It allows calculating $a_n$ directly, without going through $a_1,\dots,a_{n-1}$. The converse problem — given an explicit formula, find the recursive definition — does not make much sense, or, rather, it is trivial. If $a_n=f(n)$, then $a_n=g(n,a_{n-1})$ where $g(x,y)=f(x)$, i.e., g does not use the second argument. In this particular case, the recursive definition $a_{n}=a_{n-1}+3\cdot4^{n-1}$ is not even simpler than the explicit formula $a_{n}=4^{n}+12$.

Edit 1: The converse problem would make sense if in $a_n=g(n,a_{n-1})$ the function g can only depend on $a_{n-1}$ but not n. This restricted form of recursion is sometimes called iteration. But in this case $a_{n}$ is not a function of $a_{n-1}$, i.e., the sequence $a_n$ cannot be defined using iteration.

Edit 2: To be more precise, every sequence $a_n$ defined by (primitive) recursion can also be defined by iteration, but the definition requires an intermediate sequence: first one defined $b_n$ by iteration and then $a_n = h(b_n)$ for some function h.

11. Re: Finding the recursive definition for an integer sequence

Originally Posted by MarkFL2
$a_{n+1}=4^{n+1}+12$

$a_{n+1}-a_{n}=4^{n+1}-4^{n}+12-12$

$a_{n+1}-a_{n}=4^{n}(4-1)$

$a_{n+1}=a_{n}+3\cdot4^{n}$
Really confused how you made these transitions. Can you help me understand?

12. Re: Finding the recursive definition for an integer sequence

Nevermind, I just realized that if you subtract 4n from 4*4n you get 3*4n

Sorry its been a while