Final answer:
To have $180,000 in 35 years in an account with an annual interest rate of 12%, compounded quarterly, one would need to deposit approximately $3,229.08 initially.
Step-by-step explanation:
To determine the lump sum that should be deposited in an account with a 12% annual interest rate, compounded quarterly, to grow to $180,000 in 35 years, we use the compound interest formula:
A = P(1 + r/n)nt
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial lump sum).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.
In this case:
- A = $180,000
- r = 12% or 0.12
- n = 4 (since the interest is compounded quarterly)
- t = 35 years
Rearranging the formula to solve for P (the principal amount), we get:
P = A / (1 + r/n)nt
Substituting the given values and calculating P gives us:
P = $180,000 / (1 + 0.12/4)4*35
Now, we calculate the denominator:
(1 + 0.12/4)4*35 = (1 + 0.03)140
Calculating this exponent on a calculator, we get:
P = $180,000 / (approximately 55.7527)
So, the initial lump sum to be deposited is approximately:
P ≈ $3,229.08