calculate the amount that will be accumulated after 20 years if $1600 is invested at the beginning of every six months at 13.5% compounded semi annually.
Do not round intermediate calculations, just round the final answer to 2 decimal places
To calculate the amount that will be accumulated after 20 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount after the specified time period
P = the principal amount (initial investment)
r = the annual interest rate (expressed as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the principal amount (P) is $1600, the annual interest rate (r) is 13.5% or 0.135 (expressed as a decimal), the interest is compounded semi-annually (n = 2), and the time period (t) is 20 years.
Let's calculate it step by step:
Step 1: Calculate the semi-annual interest rate:
The annual interest rate is 13.5%, so the semi-annual interest rate will be half of that:
Semi-annual interest rate = 0.135 / 2 = 0.0675
Step 2: Calculate the total number of compounding periods:
Since the interest is compounded semi-annually, the total number of compounding periods will be twice the number of years:
Total compounding periods = 2 * 20 = 40
Step 3: Calculate the accumulated amount using the formula:
A = P(1 + r/n)^(nt)
A = 1600(1 + 0.0675/2)^(2*20)
Let's calculate the expression inside the parentheses and raise it to the power of (2*20):
(1 + 0.0675/2) = 1.03375
(1.03375)^(2*20) = 1.03375^40 ≈ 3.4651
Now, we can substitute it back into the equation:
A = 1600 * 3.4651
Calculating this, we get:
A ≈ $5,544.16
So after 20 years, the amount that will be accumulated will be approximately $5,544.16.