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If [tex]$2,000[/tex] were invested at the end of every three months, how long would it take to accumulate [tex]$475,000[/tex] at 10% compounded quarterly? (Round up to the nearest whole year.)

A) 20 years
B) 27 years
C) 60 years
D) 79 years
E) 14 years

Answer :

To solve the problem of finding out how long it will take to accumulate [tex]$475,000 when investing $[/tex]2,000 at the end of every three months at a 10% annual interest rate compounded quarterly, you can follow these steps:

1. Understand the Interest Compounding:
- Since the interest is compounded quarterly, the annual interest rate of 10% is divided by 4 (for the four quarters in a year). This gives you a quarterly interest rate of [tex]\( \frac{10}{4} = 2.5\%\)[/tex] or 0.025 in decimal.

2. Identify the Formula Used:
- The formula to calculate the future value of a series of regular investments (also known as an annuity) compounded at regular intervals is:
[tex]\[
FV = Pmt \times \left( \frac{(1 + r)^n - 1}{r} \right)
\][/tex]
Where:
- [tex]\( FV \)[/tex] is the future value of the investment (which is [tex]$475,000 in this case).
- \( Pmt \) is the payment amount per period ($[/tex]2,000 every quarter).
- [tex]\( r \)[/tex] is the interest rate per period (0.025 quarterly).
- [tex]\( n \)[/tex] is the number of periods (how many quarters it will take).

3. Rearrange the Formula to Solve for [tex]\( n \)[/tex]:
- To find [tex]\( n \)[/tex], the number of periods, the formula becomes:
[tex]\[
n = \frac{\log\left(\frac{FV \times r}{Pmt} + 1\right)}{\log(1 + r)}
\][/tex]
This rearrangement helps in solving [tex]\( n \)[/tex] directly.

4. Calculate the Number of Quarters:
- Plug in the values:
- [tex]\( FV = 475,000 \)[/tex]
- [tex]\( Pmt = 2,000 \)[/tex]
- [tex]\( r = 0.025 \)[/tex]
- After calculating using the rearranged formula, you find the number of quarters required.

5. Convert Quarters to Years:
- Since there are 4 quarters in a year, you divide the number of quarters by 4 to convert it into years.

6. Round up to the Nearest Whole Year:
- Even if the calculation results in a fraction of a year, you need to round up to the nearest whole year to ensure the target amount is reached.

By following these steps, you find that it will take approximately 78.44 quarters, which when converted and rounded up, equals 20 years. Thus, the answer is:

A) 20 years

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