Lets label them first, <span><span>bn</span>=</span> end of year balance
<span><span>b<span>n−1</span></span>=</span><span> start of year balance
</span>then,
<span><span>bn</span>=1.05<span>b<span>n−1</span></span>−10</span>
is correct formula to use. Note that the initial deposit amount is <span><span>b0</span>=1000 </span>using your formula, after 1 year, the balance would be,
<span><span>b1</span>=1.05<span>b<span>1−1</span></span>−10</span>
<span><span>b1</span>=1.05<span>b0</span>−10</span>
<span><span>b1</span>=1.05×1000−10</span>
<span><span><span>b1</span>=1040 </span></span>
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