Problem 22 - Algebra 2 and Trigonometry

Question

When \(\$ 100\) are invested at 4\(\%\) interest compounded quarterly, the value of the investment after 3 years is 100\((1+0.01)^{12} .\) Use the binomial theorem to express the value in sigma notation.

Step-by-Step Solution

Answer

100 times the sum from \(k=0\) to \(12\) of \(\binom{12}{k}(0.01)^k\).

Step 1: Understand the Binomial Theorem

The binomial theorem states that \[(1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k,\]where \( \binom{n}{k} \) is the binomial coefficient. We will use this formula to expand \((1 + 0.01)^{12}\).

Step 2: Identify Parameters for Expansion

Set \( x = 0.01 \) and \( n = 12 \). This corresponds to the expression you have \((1+0.01)^{12}\).

Step 3: Apply Binomial Expansion

Using the binomial theorem, we can write \[(1 + 0.01)^{12} = \sum_{k=0}^{12} \binom{12}{k} (0.01)^k.\]\(\binom{12}{k}\) gives us the binomial coefficients for every term in the expansion.

Step 4: Express in Sigma Notation

Combine all the terms to express the entire expansion in sigma notation:\[100 \sum_{k=0}^{12} \binom{12}{k} (0.01)^k.\]This is the expression of the investment value using the binomial theorem.

Key Concepts

Compound Interest

Compound interest is a way of calculating interest that takes into account the accumulated interest from previous periods. It's different from simple interest, where interest is only calculated on the initial amount. The key elements of compound interest are:

• Principal Amount: The initial sum of money invested or borrowed.
• Interest Rate: The percentage at which the principal grows each period.
• Compounding Frequency: How often the interest is applied (e.g., quarterly, annually).
• Time: The duration the money is invested or borrowed.

To calculate compound interest, the formula is: \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (initial investment).
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the number of years the money is invested or borrowed.

In the exercise, for example, \( \$100 \) is invested at a 4% annual interest rate compounded quarterly. Over three years, the interest compounds 12 times, making our compounding intervals very important in calculating the future value.

Sigma Notation

Sigma notation, also known as summation notation, is a compact way to represent the sum of a series. It uses the Greek letter sigma (\(\Sigma\)) to depict this process, which allows mathematicians to handle large series more efficiently. Comprehending sigma notation means understanding:

• Index of Summation: Usually represented by a variable, like \( k \), indicating the starting point of summation.
• Upper and Lower Bounds: These tell you where to start and end your summation. For example, \( k=0 \) to \( n \) means you start summing from 0 until \( n \).
• Summand: The actual formula or expression you evaluate and sum up for each value of \( k \).

In our context, using the binomial expansion, sigma notation helps us efficiently write the entire expression for compound interest. Instead of writing all separate terms, the sigma notation condenses them into: \[ 100 \sum_{k=0}^{12} \binom{12}{k} (0.01)^k. \]This notation simplifies the representation, enabling quick calculations especially helpful for expressions involving many terms.

Binomial Coefficients

Binomial coefficients arise in the expansion of expressions raised to a power, such as \((1 + x)^n\). These coefficients provide the weight of each term in the binomial expansion and are denoted as \( \binom{n}{k} \). Understanding binomial coefficients involves knowing:

• How to Calculate: \( \binom{n}{k} = \frac{n!}{k! (n-k)!} \)
• Factorial Notation: \( n! \) represents the product of all positive integers up to \( n \).
• Role in Binomial Theorem: Each \( k \) term in the expansion has a specific coefficient depending on \( n \) and \( k \).

In the binomial expansion of \((1 + 0.01)^{12}\), the coefficients \( \binom{12}{k} \) for each term determine the number of ways to choose \( k \) elements from a set of \( 12 \) elements without regard to order. The coefficients combine with powers of \( x \) to weigh each term's contribution to the total. By evaluating through sigma notation, you can maintain a systematic way to calculate the compounded total, as shown in the complete sigma expression: \( 100 \sum_{k=0}^{12} \binom{12}{k} (0.01)^k \). This illustrates the power and utility of binomial coefficients in compounding calculations, streamlining complex multipart processes.