Problem 71 - Algebra and Trigonometry

Question

Compound Interest 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

First six terms: 2004, 2008, 2012, 2016, 2020, 2024; Amount after 3 years: ~$2125.

Step 1: Understand the problem

We have Julio depositing $2000 in an account with an interest rate of 2.4% per year, compounded monthly. We want to find the first six terms and the amount after 3 years.

Step 2: Calculate the monthly interest rate

The annual interest rate is 2.4%, so the monthly interest rate is \( \frac{0.024}{12} = 0.002 \).

Step 3: Calculate each term of the sequence

For each month \(n\), use the formula \(A_n = 2000\left(1 + 0.002\right)^n\). We'll compute this for the first six months:

Step 4: First term (n=1)

Calculate \(A_1 = 2000\times(1.002)^1 = 2004\).

Step 5: Second term (n=2)

Calculate \(A_2 = 2000\times(1.002)^2 \approx 2008.004\).

Step 6: Third term (n=3)

Calculate \(A_3 = 2000\times(1.002)^3 \approx 2012.020\).

Step 7: Fourth term (n=4)

Calculate \(A_4 = 2000\times(1.002)^4 \approx 2016.048\).

Step 8: Fifth term (n=5)

Calculate \(A_5 = 2000\times(1.002)^5 \approx 2020.088\).

Step 9: Sixth term (n=6)

Calculate \(A_6 = 2000\times(1.002)^6 \approx 2024.140\).

Step 10: Calculate amount after 3 years (n=36)

To find the amount after 3 years, note that 3 years equals 36 months. Calculate \(A_{36} = 2000\times(1.002)^{36} \approx 2124.977\).

Step 11: Conclusion

The first six terms are approximately $2004, $2008, $2012, $2016, $2020, and $2024. After 3 years, the amount will be approximately $2125.

Key Concepts

Savings Account

A savings account is a secure place to store your money. It earns interest, which increases the amount over time. Typically offered by banks, savings accounts are a low-risk way to achieve financial growth.
One of the main features of a savings account is its interest-earning capability. This means while your money sits in the account, it starts to grow—not just from what you add, but because the bank is paying you for holding your money with them!
Here's what you generally find in a savings account:

• Initial deposit: The amount you start with, like Julio’s $2000 in the exercise.
• Interest rate: This determines how much extra money you'll earn.
• Compound frequency: This is how often the bank calculates and adds interest to your account balance. For Julio, this is monthly.

Ultimately, a savings account can help achieve financial goals, whether saving for education, emergencies, or a dream vacation.

Interest Rate

Interest rates tell us how quickly our money grows over time. They’re often expressed as a percentage and can be thought of as the cost of borrowing money—or conversely, the reward for saving it.
In Julio's case, he has a 2.4% interest rate. This means for every hundred dollars in the account over a year, it earns \(2.40.
But wait! Because Julio’s interest is compounded monthly, the monthly interest rate needs to be calculated. We do this by dividing the annual rate by 12 since there are 12 months in a year:
\[\text{Monthly interest rate} = \frac{2.4\%}{12} = 0.2\% = 0.002\]
Knowing the monthly interest rate helps us see how much Julio's \)2000 will grow each month, allowing us to calculate the compound interest accurately.

Sequence Formula

The sequence formula provides a systematic way to calculate the amount in the account over time. It uses the initial deposit, the compound interest rate, and the time passed.
For Julio, the sequence formula is \(A_{n}=2000\left(1+0.002\right)^{n}\). This formula uses:

• \(2000\): The initial deposit amount.
• \(0.002\): The monthly interest rate.
• \(n\): The number of months.

This formula shows how to find the account balance at any given month. Here's how it works:

• Start with the initial amount \(2000\).
• Increase by the growth factor \((1 + 0.002)\) for each month \(n\).
• Compute \((1.002)^{n}\) to find how each month's interest compounds from the previous $2000 plus interest.

Calculating this helps determine how the account balance evolves over time. It's a precise way to predict earnings from compound interest.

Monthly Compounding

Monthly compounding means that interest is calculated and added to the account balance every month. This frequent compounding can significantly boost the amount of money in a savings account over time.
With monthly compounding, the interest you've earned so far starts to earn even more interest. This creates a snowball effect where your balance grows faster because you earn interest on top of interest.
In Julio’s savings account, since the annual interest rate is 2.4%, we divide this rate by 12 to get the monthly rate. Then, every month, the formula \(A_{n} = 2000\left(1+0.002\right)^{n}\) is used to calculate the growing balance.
Monthly compounding is particularly powerful because:

• It accelerates the growth of savings compared to annual compounding.
• It allows for earlier gains as interest is applied more frequently.
• It magnifies the effect of interest over a long period, like the 36 months or 3 years in Julio's scenario.

Thus, this strategy allows Julio's savings account to benefit from compounded growth, increasing the total in his account gradually yet steadily.