Using the compound intrest formula's A=P(1+r/n)^nt and A=Pe^rt to solve the problem given. Round the answer to the nearest cent. I have to find the accumulated value of and investment of $20,000 for 6 years at an intrest rate of 4.5% compounded semiannually,quarterly,monthly, and contiuously.
compounded semi-annually
A = 20 000(1.0225)^12
= 26 121.00
compounded quarterly
... you do it
compounded monthly
.... you do it
compounded continuously
A = 20 000(e^(.045(6))
= 26 199.29
To find the accumulated value of the investment using compound interest, we can use the given formulas:
1. For compounding semiannually:
A = P(1 + r/n)^(nt)
Where:
A = Accumulated value
P = Principal amount (initial investment)
r = Interest rate (in decimal form)
n = Number of compounding periods per year
t = Time (in years)
Given:
P = $20,000
r = 4.5% = 0.045 (converted to decimal)
t = 6 years
For semiannual compounding:
n = 2 (since it's compounded semiannually - twice per year)
Now we can substitute the given values into the formula and calculate the accumulated value for the investment compounded semiannually:
A = 20000(1 + 0.045/2)^(2*6)
A = 20000(1 + 0.0225)^(12)
A ≈ $27,043.30 (rounded to the nearest cent)
To calculate the accumulated value for quarterly compounding, monthly compounding, and continuous compounding, we can use the same formula but adjust the value of n:
2. Quarterly Compounding:
n = 4 (since it's compounded quarterly - four times per year)
A = 20000(1 + 0.045/4)^(4*6)
A = 20000(1 + 0.01125)^(24)
A ≈ $27,109.11 (rounded to the nearest cent)
3. Monthly Compounding:
n = 12 (since it's compounded monthly - twelve times per year)
A = 20000(1 + 0.045/12)^(12*6)
A = 20000(1 + 0.00375)^(72)
A ≈ $27,163.05 (rounded to the nearest cent)
4. Continuous Compounding:
We can use the formula A = Pe^(rt):
A = 20000e^(0.045*6)
A ≈ $27,171.21 (rounded to the nearest cent)
Therefore, the accumulated values for the investment of $20,000 for 6 years at an interest rate of 4.5%, compounded semiannually, quarterly, monthly, and continuously are approximately:
- Semiannually: $27,043.30
- Quarterly: $27,109.11
- Monthly: $27,163.05
- Continuously: $27,171.21