# How do i find an explicit formula for this recursive linear function?

• Nov 25th 2010, 03:26 AM
oleholeh
How do i find an explicit formula for this recursive linear function?
$a_n = b+ca_{n-1}$ for constants b and c
• Nov 25th 2010, 04:11 AM
chisigma
We can start with the general case of a first order linear recursive equation...

$a_{n}= \alpha_{n}\ a_{n-1} + \beta_{n} , a_{0}=A$ (1)

With a little of patience You find that the solution of (1) is...

$a_{n}= p_{n} (A + \frac{\beta_{0}}{p_{1}} + \frac{\beta_{1}}{p_{2}} + ... + \frac{\beta_{n-1}}{p_{n}})$ (2)

... where...

$p_{n}= \alpha_{0}\ \alpha_{1}... \alpha_{n-1}$ (3)

If $\alpha_{n}=b$ and $\beta_{n}= c$ the (2) becomes...

$a_{n}= b^{n}\ (A + c\ \frac{1-b^{n}}{1-b})$ (4)

Kind regards

$\chi$ $\sigma$