Problem 122 - Precalculus with Limits

Question

A deposit of \( \$10,000 \) is made in an account that earns \( 8.5\% \) interest compounded quarterly. The balance in the account after \( n \) quarters is given by \( A_n = 10,000 \left(1 + \dfrac{0.085}{4} \right)^n , n = 1, 2, 3, . . . . \) (a) Write the first eight terms of the sequence. (b) Find the balance in the account after 10 years by computing the 40th term of the sequence. (c) Is the balance after 20 years twice the balance after 10 years? Explain.

Step-by-Step Solution

Answer

For part (a), the first eight terms of the sequence are calculated using the formula \( A_n = 10,000 \left(1 + \dfrac{0.085}{4} \right)^n \). Part (b) requires using the formula with \( n = 40\) to find the balance after 10 years, and for part (c), it's found that the balance after 20 years is not twice as much as that after 10 years.

Step 1: Part (a) - Calculate first eight terms

Use the formula \( A_n = 10,000 \left(1 + \dfrac{0.085}{4} \right)^n \) for \( n = 1, 2, 3, . . . 8 \). Calculate the first eight terms.

Step 2: Part (b) - Calculate balance after 10 years

To find out the balance after 10 years, you need to compute the 40th term of the sequence, because each term represents a quarter and there are 4 quarters in a year. So, use the formula \( A_n = 10,000 \left(1 + \dfrac{0.085}{4} \right)^n \) for \( n = 40\). Calculate the result.

Step 3: Part (c) - Compare balance after 10 and 20 years

Now, to find out if the balance in the account after 20 years is twice that of 10 years, compute the 80th term using the same formula \( A_n = 10,000 \left(1 + \dfrac{0.085}{4} \right)^n \) for \( n = 80\). Compare this result with double the balance after 10 years.

Key Concepts

Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In financial mathematics, geometric sequences often appear in the context of compound interest. In the exercise provided, each term of the sequence represents the account balance at a specific quarter. The multiplication is represented by the factor \(1 + \frac{0.085}{4}\), which is the common ratio here, derived from the interest rate divided by the number of compounding periods per year.

You can think of geometric sequences like a growing staircase, where each step is proportionally larger than the previous. In the context of the exercise, the stair steps are the quarterly balances. As you go up each step, the account balance increases by a factor determined by the compounded interest.

It's important to understand that geometric sequences are crucial for predicting future values in scenarios involving exponential growth, like investments that accrue interest over time.

Exponential Growth

Exponential growth refers to a process that increases quantity over time, where the growth is proportional to the current quantity. In simple terms, the rate of growth accelerates over time, becoming faster as the quantity grows. This is a key principle in understanding how money grows in an interest-earning account.

The formula provided in the exercise, \(A_n = 10,000 \left(1 + \frac{0.085}{4} \right)^n\), illustrates this perfectly. It shows that as time progresses, the balance increases not by a constant amount, but by a constant percentage of its current value.

This is the power of exponential growth: it begins slowly but becomes very rapid over longer periods. In the original exercise, the account balance doesn't just increase linearly; it increases more and more rapidly as the quarters pass because of the perpetual effect of compounding interest. Hence, after 20 years, the balance is not twice the amount after 10 years; it is significantly more due to the exponential nature of compound interest.

Financial Mathematics

Financial mathematics is the study of applying mathematical methods to solve problems in finance, including calculations involving interest, investments, and financial forecasts. The exercise you've seen is a classic example involving the computation of compound interest, which is a fundamental concept in this field.

Highlights of financial mathematics include understanding compound interest, time value of money, and risk assessment. Compound interest is where you earn interest on your initial principal and also on the accumulated interest from previous periods.

• **Principal**: The initial sum of money deposited or invested.
• **Quarterly compounding**: Here, interest is calculated and added to the balance 4 times a year.
• **Interest rate**: The percentage at which the investment grows per period.

By applying these concepts through formulas like the one given, you can predict account balances over different periods, giving insight into how long-term savings can be optimized. Financial mathematics equips you with the tools to make informed decisions about investments and understand how various factors affect the growth of your assets.