Problem 117 - Precalculus with Limits

Question

A deposit of \( \$100 \) is made at the beginning of each month in an account that pays \( 6\% \) interest, compounded monthly. The balance \( A \) in the account at the end of \( 5 \) years is \( A = 100 \left(1 + \dfrac{0.06}{12}\right)^1 + \cdots + 100\left(1 + \dfrac{0.06}{12}\right)^{60} \) Find \( A \).

Step-by-Step Solution

Answer

After evaluating, we find the account balance \(A\) will be approximately \$8235.05 after 5 years.

Step 1: Identify the type of series

The series provided in the problem statement is a decreasing geometric series with initial term 100, common ratio \(1 + \dfrac{0.06}{12}\), and 60 terms.

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

The sum \(S\) of a geometric series is given by the formula \(S = a \cdot \dfrac{1 - (r^n)}{1 - r}\), where \(a\) is the initial term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 3: Substitute the values and compute the sum

Substitute \(a = 100\), \(r = 1 + \dfrac{0.06}{12}\) and \(n = 60\) into the formula to compute the sum: \(A = 100 \cdot \dfrac{1 - \left(1 + \dfrac{0.06}{12}\right)^{-60}}{\dfrac{0.06}{12}}\).

Key Concepts

Understanding Compound Interest

Compound interest is a key concept when managing savings and investments. It refers to the process of earning interest not only on the initial principal but also on the accumulated interest from prior periods. This concept is particularly powerful when:

• Interest is compounded over smaller time increments like monthly or daily.
• You make regular contributions to the principal, such as monthly deposits.

When calculating compound interest, the interest rate is divided according to the compounding period. In our exercise, this means a monthly rate because the annual rate of 6% is divided by 12. This smaller incremental growth allows the investment to grow faster compared to simple interest.
In practical terms, the impact of compound interest grows over time. More frequent compounding leads to more accumulated interest. Therefore, it's essential to visualize how the interest affects both short-term and long-term growth.

The Power of Monthly Deposits

Monthly deposits are an effective way to build savings steadily over time. By regularly adding a fixed amount, here $100, to an account, you are continually increasing the principal on which interest is earned. This regular contribution becomes especially beneficial with compound interest, as each monthly deposit begins compounding the next month it is added.
These deposits foster disciplined savings habits. They also override the need for substantial initial contributions.

• Consistency: Regular monthly deposits can also take advantage of dollar-cost averaging in financial markets.
• Discipline: Automating deposits makes it easier to prioritize saving.

This method exemplifies how small, steady inputs can lead to significant financial growth over time, especially in an account earning compound interest.

Calculating the Sum of a Geometric Series

The challenge in our exercise is calculating the total amount accumulated after repeated deposits in an account with compound interest, modeled with a geometric series.
A geometric series is a set of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. In this case:

• Initial term: $100 (monthly deposit).
• Common ratio: \(1 + \frac{0.06}{12}\) (monthly interest factor).
• Number of terms: 60 (reflecting 5 years of monthly deposits).

Using the formula for the sum of a geometric series \(S = a \cdot \frac{1 - (r^n)}{1 - r}\), you can quantify how much money accumulates. Each deposit grows individually based on how long it has been in the account. This demonstrates the effect of compound growth across multiple deposits, highlighting the mathematical beauty and financial power of geometric series.