Help with recurrence relation and interest rates

Question

how would I create a recurrence relation for the amount of money in a saving account after $n$ months $a_n$ , if the interest rate is $.5\%$ interest per month and initially the account has $\$1000$?

what would the formula be for $a_n$?

Answer 1

After the first month, you will have $\color{blue}{a_1 = 1000 + 0.005\times1000 = \$1005}$

which is the initial balance + interest money.

Now after another month you will have

$a_2 = 1005 + 0.005 \times 1005$ which is the same as

$$\color{purple}{a_2 = a_1 + 0.005 \times a_1}$$

And so in general, you would have

$$\color{red}{a_n = a_{n-1} + 0.005 \times a_{n-1}}$$

I am assuming that $.5\%$ means $0.5\%$ which is $\frac{0.5}{100} = 0.005$