Assume you invest \$250 at the end of each year for 10 years at an annual interest rate of $r$. The amount of money in your account after 10 years is given by $A\left(r\right)=\frac{250({(1+r)}^{10}-1)}{r}$. Assume your goal is to have \$3500 in your account after 10 years.


b. Use a calculator to estimate the interest rate required to reach your financial goal.

Verified step by step guidance

1. Identify the equation for the amount in the account after 10 years: \( A(r) = \frac{250((1+r)^{10} - 1)}{r} \).
2. Set the equation equal to your financial goal: \( \frac{250((1+r)^{10} - 1)}{r} = 3500 \).
3. Rearrange the equation to isolate terms involving \( r \): \( 250((1+r)^{10} - 1) = 3500r \).
4. Simplify and solve for \( r \) using numerical methods or a calculator: \( (1+r)^{10} - 1 = \frac{3500r}{250} \).
5. Use a calculator to estimate \( r \) by trying different values until the left side of the equation is approximately equal to the right side.

Key Concepts

Future Value of an Annuity

The future value of an annuity formula calculates the total amount of money accumulated after making regular investments over time, considering a specific interest rate. In this case, the formula A(r) = 250 * ((1 + r)^{10} - 1) / r represents the future value of investing $250 at the end of each year for 10 years at an interest rate r. Understanding this concept is crucial for determining how much money will be available after the investment period.

Interest Rate

The interest rate is the percentage at which money grows over time when invested or borrowed. In this problem, the interest rate r is a variable that affects the future value of the annuity. Estimating the interest rate required to reach a specific financial goal, such as $3500, involves solving for r in the future value formula, which can be complex and may require numerical methods or financial calculators.

Numerical Methods

Numerical methods are techniques used to approximate solutions for mathematical problems that cannot be solved analytically. In this context, since the equation for future value involves the variable r in a non-linear way, numerical methods such as the Newton-Raphson method or bisection method can be employed to estimate the interest rate needed to achieve the desired future value. These methods are essential for finding solutions in real-world financial scenarios.