Use the appropriate formula to find the future value (in ) of
deposited at the beginning of every six months, for 17 years if a bank
pays interest, compounded semiannually. (Round your answers to the
nearest cent.)

Ask by Dickson Hill. in the United States

Mar 21,2025

Question

Use the appropriate formula to find the future value (in ) of
deposited at the beginning of every six months, for 17 years if a bank
pays interest, compounded semiannually. (Round your answers to the
nearest cent.)

Ask by Dickson Hill. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

We are given that a deposit of $$ \$300 $$ is made at the beginning of every six‐month period over 17 years. The bank pays an annual interest rate of $$ 4\% $$ compounded semiannually. This means:

- The interest rate per six‐month period is  
  $$
  r = \frac{4\%}{2} = 0.02.
  $$
- The number of six‐month periods in 17 years is  
  $$
  n = 17 \times 2 = 34.
  $$

Since the deposits are made at the beginning of each period, we are dealing with an annuity due. The future value of an annuity due is given by

$$
FV = R \times \frac{(1 + r)^n - 1}{r} \times (1+r),
$$
  
where
- $$ R = 300 $$ (deposit per period),
- $$ r = 0.02 $$,
- $$ n = 34 $$.

Substitute the values into the formula:

$$
FV = 300 \times \frac{(1.02)^{34} - 1}{0.02} \times 1.02.
$$

**Step 1. Compute $$(1.02)^{34}$$:**

Using the exponential relation,
$$
(1.02)^{34} = \exp(34 \ln 1.02).
$$
A calculation shows that
$$
(1.02)^{34} \approx 1.96048.
$$

**Step 2. Evaluate the fraction:**

$$
\frac{(1.02)^{34} - 1}{0.02} = \frac{1.96048 - 1}{0.02} = \frac{0.96048}{0.02} \approx 48.024.
$$

**Step 3. Multiply by $$ R $$ and the annuity due adjustment:**

$$
FV = 300 \times 48.024 \times 1.02.
$$

First, multiply $$ 300 \times 48.024 $$:

$$
300 \times 48.024 \approx 14407.2.
$$

Then, multiply by $$ 1.02 $$:

$$
14407.2 \times 1.02 \approx 14695.34.
$$

Thus, the future value of the account is approximately

$$
\boxed{\$14,\!695.34}.
$$
The future value of the account is approximately \$14,695.34.

Use the appropriate formula to find the future value (in ) of
deposited at the beginning of every six months, for 17 years if a bank
pays interest, compounded semiannually. (Round your answers to the
nearest cent.)

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Use the appropriate formula to find the future value (in ) of deposited at the beginning of every six months, for 17 years if a bank pays interest, compounded semiannually. (Round your answers to the nearest cent.)

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Latest Economics Questions

Use the appropriate formula to find the future value (in ) of
deposited at the beginning of every six months, for 17 years if a bank
pays interest, compounded semiannually. (Round your answers to the
nearest cent.)