Problem 69 - Precalculus: Mathematical for Calculus

Question

Julio deposits \(\$ 2000\) in a savings account that pays \(2.4 \%\) interest per year compounded monthly. The amount in the account after \(n\) months is given by the sequence $$A_{n}=2000\left(1+\frac{0.024}{12}\right)^{n}$$ (a) Find the first six terms of the sequence. (b) Find the amount in the account after 3 years.

Step-by-Step Solution

Answer

The first six terms are \(2004, 2008.008, 2012.016, 2016.024, 2020.032, 2024.04\). After 3 years, the amount is approximately \(2149.45\).

Step 1: Understand the formula

The formula for the amount in the account after \( n \) months is given by \( A_n = 2000 \left(1 + \frac{0.024}{12}\right)^n \). Here, \( 2000 \) is the initial deposit, \( 0.024 \) is the annual interest rate, and \( 12 \) represents monthly compounding.

Step 2: Calculate the monthly interest rate

First, compute the rate per month. Divide the annual rate by 12: \( \frac{0.024}{12} = 0.002 \). Now the formula simplifies to \( A_n = 2000 \times (1 + 0.002)^n \).

Step 3: Compute the first-term of the sequence

Substitute \( n = 1 \) into the equation: \( A_1 = 2000 \times (1 + 0.002)^1 = 2000 \times 1.002 \approx 2004 \).

Step 4: Compute the second-term of the sequence

Substitute \( n = 2 \) into the equation: \( A_2 = 2000 \times (1 + 0.002)^2 = 2000 \times 1.004 \approx 2008.008 \).

Step 5: Compute the third-term of the sequence

Substitute \( n = 3 \) into the equation: \( A_3 = 2000 \times (1 + 0.002)^3 = 2000 \times 1.006 \approx 2012.016 \).

Step 6: Compute the fourth-term of the sequence

Substitute \( n = 4 \) into the equation: \( A_4 = 2000 \times (1 + 0.002)^4 = 2000 \times 1.008 \approx 2016.024 \).

Step 7: Compute the fifth-term of the sequence

Substitute \( n = 5 \) into the equation: \( A_5 = 2000 \times (1 + 0.002)^5 = 2000 \times 1.010 \approx 2020.032 \).

Step 8: Compute the sixth-term of the sequence

Substitute \( n = 6 \) into the equation: \( A_6 = 2000 \times (1 + 0.002)^6 = 2000 \times 1.012 \approx 2024.04 \).

Step 9: Calculate the amount after 3 years

Three years imply \( 12 \times 3 = 36 \) months. Substitute \( n = 36 \) into the equation: \( A_{36} = 2000 \times (1 + 0.002)^{36} = 2000 \times 1.074726 \approx 2149.45 \).

Key Concepts

sequence and series

A sequence is an ordered list of numbers, where each number in the list is called a term. In financial contexts, sequences often represent the growth of an investment or account balance over time. Here, Julio's savings account balance is modeled by a sequence that shows how his deposit grows each month.

The formula given in the exercise to calculate the sequence is:

• \( A_n = 2000\left(1 + \frac{0.024}{12}\right)^{n} \)

This represents the amount of money in the account after \( n \) months. Notice that the sequence includes compound interest, which accumulates over time.

A series, on the other hand, is the sum of terms in a sequence. Although the exercise does not specifically ask for the sum, it is crucial to understand that over time, the sequence of Julio’s account balances can be thought of as accumulating wealth through the series of increments.

monthly compounding

Monthly compounding is a frequent compounding interval where interest is calculated and added to the principal at the end of each month. It is used in this exercise to show how Julio's deposit grows throughout the months.

In Julio's example, the annual interest rate is divided by 12 to find the monthly interest rate, as there are 12 months in a year. Thus, the equation representing monthly compounding becomes:

• \( A_n = 2000\left(1 + \frac{0.024}{12}\right)^{n} \)
• Which simplifies to \( A_n = 2000 \times (1 + 0.002)^n \)

This method effectively shows how every month, a small portion of the annual interest rate is added to the principal balance, affecting subsequent interest calculations, leading to exponential growth of the investment. Monthly compounding benefits savers by adding interest to the account sooner and more frequently.

interest rate calculation

Interest rate calculation is a fundamental aspect of understanding how investments and loans grow over time. In Julio's case, a 2.4% annual interest rate is used, but it is essential to break it down for monthly compounding.

The step-by-step process involves dividing the annual interest rate by the number of compounding periods in a year to get the rate per period. For monthly compounding, this calculation is:

• \( \text{Monthly Rate} = \frac{0.024}{12} = 0.002 \)

This monthly rate reflects the interest accrued each month on the existing balance. By applying this rate in the formula, the Exercise shows the power of compounding interest and how regular interest accumulation advances the total amount in Julio's saving account with time.

Understanding this calculation helps visualize how different interest rates and compounding frequencies will impact the growth of an investment or the total cost of a loan over its duration.