Rita has deposited $1000 in a bank account. at the end of each year, the bank posts interest to her account in the amount of 5% of the balance, but then takes out a $10 annual fee.
would i write it
bn = 1.05bn - 10
or
bn = 1.05b(n-1) - 10
the n is subscript

5% of $1000 is $50 bringing up her balance to $1050 minus the $10 fee at the end the year making her balance $1040.
For year 2 it will be $1082
For year 3 it will be $1126.10
For year 4 it will be $1172.41
For year 5 it will be $1231.03

Lets label them first,
bn= end of year balance
bn−1= start of year balance

then, bn=1.05bn−1−10 is correct formula to use.
Note that the initial deposit amount is b0=1000
using your formula, after 1 year, the balance would be,
b1=1.05b1−1−10
b1=1.05b0−10
b1=1.05×1000−10
b1=1040