finding recursive formula and show it converges to a limit Suppose we are playing cards and we start with $1000$ dollars. Every hour we lose $\frac{1}{2}$ of our money and then we buy another $100$ dollars. I am trying to find $x_n$ for the amount of money the player has after $n$ hours.

I think we can just take $x_n = \frac{x_{n-1}}{2} + 100 $

An so, let $L = \lim x_n$. Then $L = \frac{L}{2} + 100 $ and so $L = 200$

Is this correct?
What you did is not wrong, but it's not complete.

What you did prove:


  If the sequence $x_n$ has a limit, then the limit is equal to $200$.


What you did not prove:


  The sequence $x_n$ has a limit.




Also, that's not what the question is asking you. The question says you need to find a formula for $x_n$, not the limit of $x_n$.
$x_n = \frac{x_{n-1}}{2} + 100 = (\frac{\frac{x_{n-2}}{2} + 100}{2}) + 100 =\frac{x_{n-2}}{4} + 100.(\frac{1}{2} + 1)= \frac{\frac{x_{n-3}}{2} + 100}{4} + 100.(\frac{1}{2} + 1)=...$

$\implies x_n = \frac{x_0}{2^n} + 100.(\frac{1}{2^{n-1}} + \frac{1}{2^{n-2}} + ... + \frac{1}{2} + 1 ) $

where $x_0 = 1000$.