Problem 26 - Algebra 2 and Trigonometry

Question

Heidi deposited \(\$ 400\) at the beginning of each year for six years in an account that paid 5\(\%\) interest. At the end of the sixth year, her first deposit had earned interest for six years and was worth 400\((1.05)^{6}\) dollars, her second deposit had earned interest for five years and was worth 400\((1.05)^{5}\) dollars, her third deposit had earned interest for four years and was worth 400\((1.05)^{4}\) dollars. This pattern continues. a. What is the value of Heidi's sixth deposit at the end of the sixth year? Express your answer as a product and as a dollar value. b. Do the values of these deposits after six years form a geometric sequence? Justify your answer. c. What is the total value of Heidi's six deposits at the end of the sixth year?

Step-by-Step Solution

Answer

a. As product: \(400(1.05)^1\), Dollar value: $420. b. Yes, it's geometric. c. Total value: $2856.84.

Step 1: Calculate Heidi's Sixth Deposit Value as a Product

The sixth deposit was made at the beginning of the sixth year, so it only earns interest for one year. Therefore, the value of this deposit at the end of the sixth year is calculated as follows:\[ 400 \times (1.05)^1 \]

Step 2: Evaluate the Sixth Deposit Dollar Value

We need to evaluate the expression from Step 1 to find the dollar value. Multiplying it out, we have:\[ 400 \times 1.05 = 420 \] So, the dollar value of the sixth deposit at the end of the sixth year is $420.

Step 3: Determine if Values Form a Geometric Sequence

For a sequence to be geometric, each term after the first should be the product of the previous term and a constant ratio. The deposits grow as follows: - First deposit: \(400(1.05)^6\) - Second deposit: \(400(1.05)^5\) - Third deposit: \(400(1.05)^4\) and so on. The ratio between consecutive deposit values is \((1.05)^{-1} =\) constant. Therefore, the sequence forms a geometric sequence.

Step 4: Calculate Total Value of All Deposits

The total value is the sum of all the deposits at the end of the sixth year:\[ 400(1.05)^6 + 400(1.05)^5 + 400(1.05)^4 + 400(1.05)^3 + 400(1.05)^2 + 400(1.05)^1 \].This is a geometric sequence where- First term \(a = 400(1.05)\) - Ratio \( r = 1.05 \).The sum of a geometric series is given by the formula:\[ S_n = a\frac{r^n - 1}{r - 1} \] where \( n = 6 \).Substitute and calculate:\[ S_6 = 400(1.05)\frac{(1.05)^6 - 1}{1.05 - 1} \]Calculating:\[ 420 \times \frac{1.3401 - 1}{0.05} = 420 \times 6.802 = 2856.84 \]Thus, the total value is approximately $2856.84.

Key Concepts

Compound Interest

Compound interest is a concept that has a profound impact on how savings can grow over time. It is the interest calculated on the initial principal, and also on the accumulated interest of previous periods. With compound interest, the key is that interest is earned on the interest itself, not just the original amount of money. This causes the account balance to grow substantially over time.
Heidi's deposits are a perfect example of using compound interest. Each year, the interest earned from the previous year is added back to the principal, making the fund grow exponentially. To see this in action, if $400 is deposited and earns 5% interest, the deposit becomes:

• First year: $400
• Second year: $400 + interest from the first year
• And so on, where each year the interest increases because it's computed on the total amount of the previous year.

Recognizing the power of compound interest is essential for anyone looking to invest for the long term.

Geometric Series

A geometric series is a series of terms that have a constant ratio between successive terms. This characteristic makes it quite straightforward to calculate sums over multiple terms. The formula for the sum of a geometric series is a powerful tool to have. It is expressed as:\[ S_n = a \frac{r^n - 1}{r - 1} \]Where:

• \( S_n \) is the sum of the series.
• \( a \) is the first term.
• \( r \) is the common ratio between terms.
• \( n \) is the number of terms.

In Heidi's case, each deposit can be viewed as a term in a geometric series. The interest compounding each year acts as the constant ratio. Recognizing this can simplify the total calculation of six deposits, as evidenced by the formula application in her total savings calculation. The sequence of values for each deposit shows that they indeed form a geometric series.

Financial Mathematics

Financial mathematics encompasses various concepts that deal with the time value of money, investments, and financial calculations. Heidi's deposits introduce a simple example of how financial mathematics applies to real-life situations—like planning and achieving savings goals. To manage finances effectively, understanding the basic calculations can help make the right investment choices. Financial calculations like those for compound interest, and geometric series allow you to project future account balances and assess the viability of investment opportunities.
These calculations involve:

• Determining the future value of deposits.
• Assessing the impact of changing interest rates on savings.
• Comparing different financial products.

Equipping yourself with skills in financial mathematics not only simplifies complex decisions but can also ensure a more secure financial future.

Sequence and Series

Sequence and series are fundamental concepts in mathematics, especially when exploring how patterns and relationships develop over time. A sequence is a set of numbers arranged in a specific order. In contrast, a series is the sum of the terms of a sequence. Both are crucial for understanding financial calculations and modeling. In the context of Heidi's savings, each deposit represents a point in a sequence. The way these grow due to compound interest forms a geometric series. Recognizing patterns in sequences and series helps make informed predictions about future outcomes. When analyzing investments or savings plans, it's important to:

• Recognize the type of sequence or series.
• Identify the pattern or rule governing it.
• Use appropriate formulas to calculate desired information.

Understanding these concepts enables better financial forecasting, crucial for both personal finance planning and broader financial analysis.