Final answer:
Larry needs an annual rate of return of approximately 0.80% to grow his $3000 investment to $3400 in 16 years, using the formula for compound interest.
Step-by-step explanation:
To solve the problem of finding the annual rate of return Larry needs to get in order to turn his $3000 investment into $3400 in 16 years, we will use the formula for compound interest:
A = P(1 + r)^n
Where:
- A is the future value of the investment/loan, including interest
- P is the principal investment amount ($3000 in this case)
- r is the annual interest rate (decimal)
- n is the number of years the money is invested (16 years in this case)
We need to find the rate r that will make A equal to $3400 after 16 years. Therefore:
3400 = 3000(1 + r)^16
Dividing both sides by 3000 gives:
1.1333 = (1 + r)^16
Now we apply the nth root to both sides to solve for (1 + r):
(1 + r) = (1.1333)^(1/16)
After calculating the 16th root, we get:
(1 + r) ≈ 1.00797
Then we subtract 1 from both sides to find r:
r ≈ 0.00797
To express this as a percentage, we multiply by 100:
Annual rate of return ≈ 0.797%
Rounded to two decimals, Larry will need an annual rate of return of 0.80% to achieve his goal of $3400 in 16 years.