We are given that an initial investment of $25,000 grew to $1,500,000 in 13 years with annual compounding. We need to find the annual interest rate.

1. Problem Description

We are given that an initial investment of $25,000 grew to$1,500,000 in 13 years with annual compounding. We need to find the annual interest rate.

2. Solution Steps

Let $P$ be the initial investment, $A$ be the final amount, $r$ be the annual interest rate (expressed as a decimal), and $n$ be the number of years. The formula for compound interest is:

$A = P(1+r)^n$

In this case, we have $P = 25000$, $A = 1500000$, and $n = 13$. We want to solve for $r$.

$1500000 = 25000(1+r)^{13}$

Divide both sides by 25000:

$\frac{1500000}{25000} = (1+r)^{13}$

$60 = (1+r)^{13}$

Take the 13th root of both sides:

$\sqrt[13]{60} = 1+r$

$60^{\frac{1}{13}} = 1+r$

$1.37615... = 1+r$

$r = 1.37615... - 1$

$r = 0.37615...$

To express this as a percentage, we multiply by 100:

$r\% = 0.37615... \times 100 = 37.615... \%$

Rounding to two decimal places, we get $37.62\%$.

3. Final Answer

37.62