Final answer:
Using the compound interest formula, it would take approximately 6.9 years for an investment to grow fivefold if it were invested at 15 percent compounded weekly.
Step-by-step explanation:
To determine approximately how many years it would take for an investment to grow fivefold with a 15 percent interest rate compounded weekly, we can use the formula for compound interest:
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 amount of money).
- 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 or borrowed for, in years.
To solve for t when the investment grows fivefold, we set A to 5 times the principal P which is initially $1. The interest rate r is 15% or 0.15, and since it's compounded weekly, n is equal to 52. Plugging in the values, we have:
5 = (1)(1 + 0.15/52)^(52t)
Now, we solve for t:
5 = (1.00288461538)^(52t)
To isolate t, we take the natural logarithm (ln) of both sides:
ln(5) = 52t * ln(1.00288461538)
t = ln(5) / (52 * ln(1.00288461538)) ≈ 6.9026 years
So, it would take approximately 6.9 years for the investment to grow fivefold when compounded weekly at a 15% interest rate.