Problem 77 - Precalculus: Mathematics 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 $$A_{n}=2000\left(1+\frac{0.024}{12}\right)^{n}$$ (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 \(2004.00, 2008.008, 2012.024, 2016.048, 2020.080, 2024.120\). After 3 years, \(A_{36} = 2144.27\).

Step 1: Understanding the Problem

The problem presents a sequence of amounts in a savings account, with monthly compounding interest. We need to calculate the sequence values for the first six months, and then find the total amount in the account after 3 years.

Step 2: Calculating Monthly Interest Rate

The given annual interest rate is \(2.4\%\). Therefore, the monthly interest rate is \(\frac{2.4}{12} = 0.2\%\) or \(0.002\) when expressed as a decimal.

Step 3: First Month Value

Using the formula, substituting \(n = 1\): \[A_1 = 2000\left(1 + 0.002\right)^{1} = 2000 \times 1.002 = 2004.00\]

Step 4: Second Month Value

Substituting \(n = 2\): \[A_2 = 2000\left(1 + 0.002\right)^{2} = 2000 \times 1.004004 = 2008.008\]

Step 5: Third Month Value

Substituting \(n = 3\): \[A_3 = 2000\left(1 + 0.002\right)^{3} = 2000 \times 1.006012008 = 2012.024\]

Step 6: Fourth Month Value

Substituting \(n = 4\): \[A_4 = 2000\left(1 + 0.002\right)^{4} = 2000 \times 1.008024016 = 2016.048\]

Step 7: Fifth Month Value

Substituting \(n = 5\): \[A_5 = 2000\left(1 + 0.002\right)^{5} = 2000 \times 1.010040024 = 2020.080\]

Step 8: Sixth Month Value

Substituting \(n = 6\): \[A_6 = 2000\left(1 + 0.002\right)^{6} = 2000 \times 1.012060048 = 2024.120\]

Step 9: Determining Number of Months in 3 Years

To find the total amount after 3 years, first calculate the number of months: \(3 \times 12 = 36\) months.

Step 10: Calculating Amount After 3 Years

Substituting \(n = 36\): \[A_{36} = 2000\left(1 + 0.002\right)^{36}\] \[A_{36} = 2000 \times 1.072135 = 2144.27\]

Key Concepts

Savings Account

A savings account is a type of financial account offered by banks and other financial institutions. It allows you to deposit money, earns interest over time, and withdraw funds when needed.
Unlike checking accounts, savings accounts are primarily used for long-term savings, providing a secure place to keep your money with the bonus of earning interest. Key features of a savings account include:

• Deposits and Withdrawals: You can add or take out money, but transfers might be limited per month.
• Interest Earnings: Money deposited earns interest, boosting your total savings over time.
• Low Risk: Funds in savings accounts are typically insured, making them a low-risk form of saving.

Investing in a savings account is a stable choice, especially when planning for future needs and building a financial buffer.

Sequence of Amounts

The sequence of amounts in a savings account represents how the account balance changes over time with each interest compounding period.
This sequence is calculated using a specific formula that factors in the initial principal, the interest rate, and the number of compounding periods.Given our text's exercise:

• The initial deposit, or principal, is \( \$2000 \).
• The interest rate is annually 2.4% but applies monthly, affecting the sequence.
When we calculate each month's balance, we see the amounts change slightly as interest adds up. Remember, this is a mathematical sequence where each term is derived from the preceding one by applying the compound interest formula. These calculated amounts help us understand the power of compound interest – allowing small, regular growth to lead to significant gains over a period.

Interest Rate

An interest rate represents the percentage at which your money grows in a savings account.
It influences how much you earn on your deposit and how quickly the account balance increases. In this context:

• The exercise states an interest rate of 2.4% per year.
• This rate is divided by twelve to convert it into a monthly rate, which becomes 0.2% per month or 0.002 in decimal form.

The interest rate's impact can be seen by calculating the sequence of amounts over time. Even a seemingly small percentage can result in noticeable growth thanks to the principle of compounding: interest is added not just to the initial amount, but to the previous interest earned as well.

Monthly Compounding

Monthly compounding refers to the process of calculating and adding interest to a savings account balance every month.
This is a common practice that leads to more frequent accumulation of interest. Here's how it works:

• Each month, the interest earned is added to the account balance.
• Future interest calculations are then based on this new total, including the previous interest accumulated.

In our exercise example, the amount in the account increases every month, not just annually. Initial calculation with monthly compounding creates greater returns over time compared to annual compounding. This benefit arises because the more often interest is compounded, the faster your account grows. With monthly compounding, even small regular interest earnings can significantly increase the total amount in a savings account over longer periods.