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

1. Problem Description

We are given that an initial investment of $25,000$ grows to $1,500,000$ in 13 years, compounded annually. 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, and $n$ be the number of years. The formula for compound interest is:

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

We are given:

$P = 25000$

$A = 1500000$

$n = 13$

We need to find $r$. Plug in the values:

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

Divide both sides by 25000:

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

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

Take the 13th root of both sides:

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

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

$1.36848... = 1 + r$

Subtract 1 from both sides:

$r = 1.36848... - 1$

$r = 0.36848...$

To express the interest rate as a percentage, multiply by 100:

$r\% = 0.36848... \times 100 = 36.848...\%$

Rounding to two decimal places, we get $36.85\%$.

3. Final Answer

36.85 %