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Problem 11

# Find the periodic payments necessary to accumulate the given amount in an annuity account. (Assume end-of-period deposits and compounding at the same intervals as deposits.) [HINT: See Quick Example 2.] $$\ \ 20,000$$ in a fund paying $$5 \%$$ per year, with monthly payments for 5 years, if the fund contains $$\ 10,000$$ at the start

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## Step 4: Apply the FVOA formula and solve for P

We have all the necessary parameters, so we can now apply the FVOA formula to solve for the periodic payment P: $$FV' = P\frac{(1+r)^n-1}{r}$$ Substitute the values of FV', r, and n into the formula: $$\10,000 = P\frac{(1+0.004167)^{60}-1}{0.004167}$$ Now solve for P using algebra: $$P = \frac{\10,000 \times 0.004167}{(1+0.004167)^{60}-1}$$ $$P = \frac{\41.67}{(1.004167)^{60}-1}$$ $$P = \frac{\41.67}{1.28323-1}$$ $$P = \frac{\41.67}{0.28323}$$ $$P \approx \147.13$$

## Step 5: Conclusion

To accumulate \$20,000 in the annuity account with a starting balance of \$10,000, periodic payments of approximately \\$147.13 are necessary for the next 5 years, with monthly deposits and compounding interest at the same intervals.

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