Given
principal initial amount
time number of years
rate interest rate as decimal
n number of periods per time to compound principal by rate
a amount to add to principal at either beginning or end of each n period
where
principal is equal to 1
time is equal to 1
rate is equal to 0.03, or 3%
n is equal to 12, or monthly
a is equal to 10
The result that derived here is 124.68033078653431, which is equivalent to initial principal 121 compounded monthly, not initial principal 1 with addition a 10 made each n month then compounded.
What are the applicable mathematical formulas to determine the current value of the accrued interest and principal where the addition a is made a) at the beginning of each period, or b) at the end of each period?
When is the interest applied to the principal during the given period?
Interest is computed at the end of each period. You can just consider each addition to be a new principal. The first $1$ becomes $1(1+r)^n$ at the end of $n$ periods. If you add $10$ at the end of the first period is becomes $10(1+r)^{n-1}$ because there is one less period to draw interest. If you add $10$ at the end of each period, you get a geometric series to sum: $10[(1+r)^{n-1}+(1+r)^{n-2}+\ldots (1+r)^1+1]=10\frac {(1+r)^n-1}{1+r}$