Find Closed Form given Recursive Formula

**Edbaseball17** · February 3rd 2009, 12:07 PM

Recursive formula:

$a_n=5*a_{n-1}$ with $a_1 = 5$

Can someone please show me how to do this? I'm studying for a test and need to see the step in order to proceed with the rest of these types of problems.

The question asks to find the next 5 terms of the sequence and then provide the Closed Form.

Thanks

**Plato** · February 3rd 2009, 12:30 PM

Originally Posted by Edbaseball17

Recursive formula:
$a_n=5*a_{n-1}$ with $a_1 = 5$

Surely you see that $a_n = 5^n$ ?

**Edbaseball17** · February 3rd 2009, 12:35 PM

Here is what I've tried and it is not working.

$a_1=5(0)$
$a_2=5(1)$

and so on.

**o_O** · February 3rd 2009, 12:41 PM

Look at your recursion again: $a_n = 5a_{n-1}$

So, we have:
$\begin{aligned} a_1 & = 5 \\ a_2 & = 5a_1 = 5 \times 5 = 5^2 \\ a_3 & = 5a_2 = 5(5^2) = 5^3 \\ a_4 & = 5a_3 = 5(5^3) = 5^4 \\ & \qquad \qquad \vdots \\ a_n & = \cdots \end{aligned}$

See how Plato's suggestion comes into play?

**Plato** · February 3rd 2009, 12:42 PM

Originally Posted by Edbaseball17

Here is what I've tried and it is not working.
$a_1=5(0)$ $a_2=5(1)$

No indeed!
You were given that $a_1=5$ then $a_{\color{blue}1}=5^{\color{blue}1}$ .
Also we know that $a_n = 5 a_{n-1}$ so $a_2=5(a_1)=5(5)=5^2$ .
Do you understand?

**Edbaseball17** · February 3rd 2009, 12:52 PM

I'm an idiot. Yea i got it now. I was substituting the sub numbers instead of using the finished product of the equation. Thanks Plato!

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