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Your 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.)

Ask by Rojas Stewart. in the United States

Mar 21,2025

Question

File Edit Format View Help
Your 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.)

Ask by Rojas Stewart. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

Let the deposit per six‐month period be $$ R = 500 $$. The interest rate per six months is

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

Since deposits are made at the beginning of each period (an annuity due) for 16 years, the number of periods is

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

The future value of an annuity due is given by

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

Substitute the values:

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

Now compute $$(1.02)^{32}$$. Using the exponential relation

$$
(1.02)^{32} = e^{32 \ln(1.02)},
$$

and noting that $$32 \ln(1.02) \approx 0.63366$$, we find

$$
(1.02)^{32} \approx e^{0.63366} \approx 1.885.
$$

Thus,

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

Now, calculate the future value:

$$
FV \approx 500 \times 44.25 \times 1.02.
$$

First, multiply $$500 \times 44.25$$:

$$
500 \times 44.25 = 22\,125.
$$

Then multiply by $$1.02$$:

$$
22\,125 \times 1.02 \approx 22\,567.50.
$$

Thus, the future value of the deposits is approximately

$$
\boxed{\$22\,567.50}.
$$
The future value of the deposits is approximately \$22,567.50.