A deposit of $\$100,000$ is made to an investment fund at the beginning of a year. On the last day of each year two dividends are awarded. The first dividend is $20\%$ of the amount in the account during that year. The second dividend is $45\%$ of the amount in the account in the previous year.
- Find a recurrence relation for $\{Pn\},$ where $P_{n}$ is the amount in the account at the end of $n$ years if no money is ever withdrawn.
- How much is in the account after $n$ years if no money has been withdrawn?
