Principal and Annuities

Question

Suppose you want to accumulate $12\,000$ in a $5 \%$ account by making a level deposit at the beginning of each of the next $9$ years. Find the required level payment. So this seems to be an annuity due problem. I know the following:

$ \displaystyle \sum_{k=1}^{n} \frac{A}{(1+i)^{k}} = \frac{A}{1+i} \left[\frac{1- \left(\frac{1}{1+i} \right)^{n}}{1- \left(\frac{1}{1+i} \right)} \right] = P$.

So in this problem, we are trying to solve for $P$? Just plug in the numbers? Or do we need to calculate the discount rate $d = i/(i+1)$ since the annuity is being payed at the beginning of the year?

Answer 1

I get $F = A \frac{(1+i)^{2n}-(1+i)^n}{i(1+i)^n}$. But when I solve for $A$, I get the level payment as being $1088.28$. But the correct answer is $1036.46$. Why is there a discrepancy?

Answer 2

The problem statement is missing the time when you want to have the 12,000. If it is at the end of the ninth year, the value of the deposit at the beginning of year n will have increased by 1.05^(10-n). So if A is the deposit you have $\displaystyle \sum_{k=1}^{n} A*1.05^{(10-k)}=12000$. Solve for A