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 $x_n$.
