Problem 121 - Algebra and Trigonometry Real Mathematics, Real People

Question

A deposit of \(5000\)dollar is made in an account that earns \(3 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by $$A_{n}=5000\left(1+\frac{0.03}{4}\right)^{n}, \quad n=1,2,3, \ldots$$ (a) Compute the first eight terms of this sequence. (b) Find the balance in this account after 10 years by computing the \(40\)th term of the sequence.

Step-by-Step Solution

Answer

The first eight terms of the sequence are respectively: $5000(1+0.03/4)^1$, $5000(1+0.03/4)^2$, $5000(1+0.03/4)^3$, $5000(1+0.03/4)^4$, $5000(1+0.03/4)^5$, $5000(1+0.03/4)^6$, $5000(1+0.03/4)^7$, $5000(1+0.03/4)^8$. After 10 years, the balance in the account is $5000(1+0.03/4)^{40}$.

Step 1: Calculate the first eight terms.

To calculate first eight terms of the sequence, substitute \(n\) with each of the first eight numbers (quarters), in the given equation \(A_{n}=5000\left(1+\frac{0.03}{4}\right)^{n}\).

Step 2: Present the first eight terms.

After calculations, the balance in the account for the first eight quarters respectively will be: $5000(1+0.03/4)^1$, $5000(1+0.03/4)^2$, $5000(1+0.03/4)^3$, $5000(1+0.03/4)^4$, $5000(1+0.03/4)^5$, $5000(1+0.03/4)^6$, $5000(1+0.03/4)^7$, $5000(1+0.03/4)^8$.

Step 3: Calculate the balance after 10 years.

To calculate the balance after 10 years i.e., 40 quarters, substitute \(n\) with 40 in the given equation \(A_{n}=5000\left(1+\frac{0.03}{4}\right)^{n}\), and compute the result.

Step 4: Present the balance after 10 years.

After calculations, the balance in the account after 10 years will be $5000(1+0.03/4)^{40}$.

Key Concepts

sequence

In mathematics, a sequence is a list of numbers arranged in a specific order, following a certain rule or pattern. These numbers are known as the terms of the sequence. In the context of compound interest, a sequence represents the balance of an account over time, calculated after each compounding period.

For example, in the exercise you're dealing with, a $5000 deposit is compounded quarterly at a rate of 3% per year. The balance in the account after each quarter is calculated using the formula \(A_{n}=5000\left(1+\frac{0.03}{4}\right)^{n}\), where \(n\) represents the number of quarters.

This means the sequence you build will show how the balance grows after each quarter.

• The first term is the balance after the first quarter.
• The second term is the balance after the second quarter, and so on.

Each of these terms shows how the interest accumulates over time, and that's what gives this pattern its meaning.

quarterly compounding

Quarterly compounding is a method of calculating interest where the interest earned is added to the principal balance four times a year, or once every three months. This is a common practice for savings accounts and investments, as it helps money grow more efficiently over time.

In our exercise, the interest rate is 3% annually, but since we're compounding quarterly, each quarter the account earns a quarter of that annual rate. That means each period earns 0.75% interest, which is calculated as \(\frac{0.03}{4}\).

With quarterly compounding, you are adding the interest earned each quarter back into the main balance.

• This helps to increase the base amount for future interest calculations.
• As a result, the interest grows on top of the interest that was earned in previous quarters, illustrating the beauty of compound interest.

This method shows how regular compounding periods can lead to exponential growth in the account balance.

exponential growth

Exponential growth describes a process where the quantity increases at a consistent rate over time, which results in a curve on a graph that gets steeper and steeper. In terms of finance, it occurs when an investment grows at a consistent percentage rate—thanks to the effects of compounding.

The formula \(A_{n}=5000\left(1+\frac{0.03}{4}\right)^{n}\) represents exponential growth. Each quarter, the interest is calculated on the new balance, including any previously earned interest, leading to growth that speeds up over time.

Thus, instead of increasing by a fixed amount, the balance increases by a percentage of the current amount, leading to quick and significant growth. This is why, after 10 years, the balance increases to more than just a simple addition of interest—it reflects the efficiency and power of exponential growth.

• This growth method is particularly advantageous for long-term investments.
• It demonstrates how small regular additions can result in substantial increases over time.

The exponential growth model is what allows investors to benefit from the 'snowball' effect in their investment accounts.