If $7000 is deposited at the end of each half year in an account that earns 6.3% compounded semiannually, after how many half years will the account contain $150,000? (Round your answer UP to the nearest half year.)
half years

We are given that:

- A deposit of $7000 is made at the end of each half year.
- The interest rate is 6.3% per annum, compounded semiannually.
- We want to determine how many half years are required for the account to reach $150,000.

The formula to use is for the future value of an ordinary annuity, which is given by:

\[
A = P \cdot \frac{(1 + i)^n - 1}{i}
\]

Where:
- $A$ is the future value (in this case, $150,000),
- $P$ is the periodic payment (in this case, $7000),
- $i$ is the interest rate per period (semiannual interest rate),
- $n$ is the number of periods (half years).

### Step 1: Calculate the semiannual interest rate
Since the annual interest rate is 6.3%, the semiannual interest rate is:

\[
i = \frac{6.3\%}{2} = 0.0315
\]

### Step 2: Substitute known values into the formula
We need to solve for $n$ when $A = 150,000$, $P = 7000$, and $i = 0.0315$:

\[
150,000 = 7000 \cdot \frac{(1 + 0.0315)^n - 1}{0.0315}
\]

### Step 3: Simplify and solve for $n$
First, divide both sides by 7000:

\[
\frac{150,000}{7000} = \frac{(1 + 0.0315)^n - 1}{0.0315}
\]

\[
21.42857 = \frac{(1.0315)^n - 1}{0.0315}
\]

Multiply both sides by 0.0315:

\[
21.42857 \times 0.0315 = (1.0315)^n - 1
\]

\[
0.675 = (1.0315)^n - 1
\]

Add 1 to both sides:

\[
1.675 = (1.0315)^n
\]

Now, take the natural logarithm (ln) of both sides:

\[
\ln(1.675) = n \cdot \ln(1.0315)
\]

\[
n = \frac{\ln(1.675)}{\ln(1.0315)}
\]

Let's calculate $n$.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can continue solving this by calculating the values manually:

1. Find $ \ln(1.675) $ and $ \ln(1.0315) $.
2. Divide these results to get $n$.
3. Round your answer up to the nearest half year.

Would you like to walk through these steps in detail? Let me know if you have any further questions.

Here are 5 related questions:
1. What is the difference between compounding annually and semiannually?
2. How does an ordinary annuity differ from an annuity due?
3. What happens if the interest rate is compounded quarterly instead of semiannually?
4. How can the future value change if you increase the deposit amount?
5. How do continuous compounding formulas differ from periodic compounding?

**Tip:** Always round up when dealing with time periods in financial calculations to ensure full accrual of interest.

If $7000 is deposited at the end of each half year in an account that earns 6.3% compounded semiannually, after how many half years will the account contain $150,000? (Round your answer UP to the nearest half year.)

Learn how to calculate the number of half years required for an account to reach $150,000 with $7000 semiannual deposits, earning 6.3% interest compounded semiannually. Includes step-by-step application of the future value of annuities formula and logarithmic equations. Suitable for Grades 11-12 and early college students.

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