Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

A deposit of \(\$ 50\) is made at the beginning of each month in an account that pays \(2 \%\) interest, compounded monthly. The balance \(A\) in the account at the end of 6 years is given by \(A=50\left(1+\frac{0.02}{12}\right)^{1}+\cdots+50\left(1+\frac{0.02}{12}\right)^{72}\) Find \(A.\)

Step-by-Step Solution

Answer

Based on the steps mentioned, the balance \(A\) in the account at the end of 6 years will approximately be \$5023.29.

Step 1: Identify the geometric series and its components

The exercise describes a geometric series with the general term \(50(1+\frac{0.02}{12})^n\), where \(n\) varies from 1 to 72. The first term \(a\) of our geometric series is therefore \(50(1+\frac{0.02}{12})\), the common ratio \(r\) is \(1+\frac{0.02}{12}\), and the number of terms \(n\) is 72.

Step 2: Apply the formula for the sum of a geometric series

The formula for the sum \(S\) of the first \(n\) terms of a geometric series is \(S = a \frac{1 - r^n}{1 - r}\). Plug in the values \(a = 50(1+\frac{0.02}{12})\), \(r = 1+\frac{0.02}{12}\), and \(n = 72\) to get \(S = 50(1+\frac{0.02}{12}) \frac{1 - (1+\frac{0.02}{12})^{72}}{1 - (1+\frac{0.02}{12})}\).

Step 3: Calculate the value of \(S\)

Now, calculate the value of \(S\). This is a regular math operation, where in our specific case you can use a calculator to get a fast result.

Key Concepts

Compound Interest

Compound interest is a powerful way to grow savings over time because it allows you to earn interest on both the initial principal and the accumulated interest from previous periods. Unlike simple interest, which is calculated only on the principal amount, compound interest increases the investment at a growing rate, amplifying growth with each compounding period.

For our exercise, the account pays interest monthly. The formula used in this context is given by:

• The monthly interest rate: \( \frac{0.02}{12} \)
• The balance grows by a factor of \( 1 + \frac{0.02}{12} \) each month.

This means every month, the interest is applied not only to the initial deposit but also to any previously earned interest, leading to a compound effect over 6 years.

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our exercise, the sequence represents the monthly deposits modified by the interest rate.

The first term of this geometric sequence is given by:

• \( a = 50(1 + \frac{0.02}{12}) \) – representing the initial deposit adjusted by one month's interest.

The common ratio \( r \) is the growth factor for the investment each month, calculated as:

• \( r = 1 + \frac{0.02}{12} \).

This structure creates a predictable pattern, where each term in the sequence increases by the factor of \( r \), driving the total balance upward.

Sum of Geometric Series

The sum of a geometric series refers to the total value when all terms of the sequence are added together. It is essential for calculating the accumulated amount after a set period with regular deposits and compound interest.

To calculate the sum \( S \) of the geometric series in our exercise, we use the formula:

• \[ S = a \frac{1 - r^n}{1 - r} \]

Where:

• \( a = 50(1 + \frac{0.02}{12}) \) is the first term,
• \( r = 1 + \frac{0.02}{12} \) is the common ratio,
• \( n = 72 \) represents the number of terms (months).

By substituting these values into the formula, you can compute the total amount in the account after 6 years, where each term represents a single month's contribution and growth.