our bank pays $$ 4 \% $$ interest, compounded semiannually. Use Table $$ 12-1 $$ to find the future value (in $$ \$ $$ ) of $$ \$ 500 $$ deposited at the beginning of every six months, for 16 years. (Round your answer to the nearest cent

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We first note that the bank pays an annual nominal interest rate of $$4\%$$ compounded semiannually. This means that the interest rate per six‐month period is

$$
i = \frac{4\%}{2} = 2\% = 0.02.
$$

Since the deposits are made at the beginning of every six months (annuity due) for $$16$$ years and there are $$2$$ periods per year, the total number of periods is

$$
n = 16 \times 2 = 32.
$$

For an annuity due, the future value $$ FV $$ is given by the formula

$$
FV = P \left(\frac{(1+i)^n - 1}{i}\right)(1+i),
$$

where $$ P = \$500 $$ is the payment per period.

Substitute the values into the formula:

$$
FV = 500 \left(\frac{(1.02)^{32} - 1}{0.02}\right)(1.02).
$$

Now, compute step by step:

1. Calculate $$(1.02)^{32}$$. Using logarithms or a calculator, we find

$$
(1.02)^{32} \approx 1.885.
$$

2. Compute the annuity factor:

$$
\frac{(1.02)^{32} - 1}{0.02} \approx \frac{1.885 - 1}{0.02} = \frac{0.885}{0.02} = 44.25.
$$

3. Adjust for the annuity due by multiplying by $$1.02$$:

$$
44.25 \times 1.02 \approx 45.135.
$$

4. Now, multiply by the periodic deposit:

$$
FV \approx 500 \times 45.135 \approx 22567.50.
$$

Thus, the future value of the deposits is approximately

$$
\$22,\!567.50.
$$
The future value of the \$500 deposits is approximately \$22,567.50.

our bank pays interest, compounded semiannually. Use Table
to find the future value (in ) of deposited at the beginning
of every six months, for 16 years. (Round your answer to the nearest cent

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our bank pays interest, compounded semiannually. Use Table to find the future value (in ) of deposited at the beginning of every six months, for 16 years. (Round your answer to the nearest cent