# Thread: Find Closed Form given Recursive Formula

1. ## Find Closed Form given Recursive Formula

Recursive formula:

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

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

2. Originally Posted by Edbaseball17
Recursive formula:
$\displaystyle a_n=5*a_{n-1}$ with $\displaystyle a_1 = 5$
Surely you see that $\displaystyle a_n = 5^n$?

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

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

and so on.

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

So, we have:
\displaystyle \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?

5. Originally Posted by Edbaseball17
Here is what I've tried and it is not working.
$\displaystyle a_1=5(0)$ $\displaystyle a_2=5(1)$
No indeed!
You were given that $\displaystyle a_1=5$ then $\displaystyle a_{\color{blue}1}=5^{\color{blue}1}$.
Also we know that $\displaystyle a_n = 5 a_{n-1}$ so $\displaystyle a_2=5(a_1)=5(5)=5^2$.
Do you understand?

6. 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!