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

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We are given that the bank pays an annual nominal interest rate of $$4\%$$ compounded semiannually. This means that every six‐month period, the interest rate is

$$
i=\frac{0.04}{2}=0.02.
$$

Since deposits of $$\$500$$ are made at the beginning of each six‐month period over 16 years, there will be

$$
n = 16\times2 = 32 \text{ deposits}.
$$

Because the deposits are made at the beginning of each period, we have an annuity due. The future value $$FV$$ of an annuity due is given by

$$
FV = R \cdot \frac{(1+i)^n - 1}{i} \cdot (1+i),
$$

where $$R = 500$$, $$i = 0.02$$, and $$n = 32$$.

Substitute the values:

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

**Step 1. Compute $$(1.02)^{32}$$:**

Using logarithms or a calculator, we find

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

**Step 2. Compute the annuity factor:**

Subtract 1:

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

Divide by $$0.02$$:

$$
\frac{0.885}{0.02} \approx 44.25.
$$

Multiply by $$1.02$$ (to convert from an ordinary annuity to an annuity due):

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

**Step 3. Multiply by the payment amount:**

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

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

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