Question

A person deposits \(\$ 10\) at the beginning of each quarter into a bank account that earns \(4 \%\) annual interest compounded quarterly (four times a year). a. Show that the interest accumulated after \(n\) quarters is \(\$ 10\left(\frac{1.01^{n+1}-1}{0.01}-n\right)\). b. Find the first eight terms of the sequence. c. How much interest has accumulated after 2 years?

Short answer

a. The formula is valid after derivation. b. Calculate each term from n=1 to n=8. c. Use formula for n=8.

Step-by-step solution

Understand the Formula for Future Value of Annuity

The future value of an annuity formula is given by \( FV = P \times \frac{(1 + r)^n - 1}{r} \) where \( P \) is the deposit per period, \( r \) is the interest rate per period, and \( n \) is the number of periods. In this case, \( P = 10 \), \( r = \frac{0.04}{4} = 0.01 \), and we need to solve for \( n \) quarters.

Calculate the Total Value Including Interest

The total value after \( n \) quarters using the formula becomes \( 10 \times \frac{(1.01)^n - 1}{0.01} \). This includes the interest as well as the principal amount.

Calculate the Total Principal Deployed

Since \( n \) deposits of \( \$10 \) are made, the total principal is simply \( 10n \).

Calculate the Accumulated Interest

Subtract the total principal from the total value to find the accumulated interest: \[ Interest = 10 \times \frac{(1.01)^n - 1}{0.01} - 10n \] Simplifying, we find: \[ Interest = 10 \times \left( \frac{(1.01)^n - 1}{0.01} - n \right) \] We notice a recurrent sequence update: \[ Interest = 10 \times \left( \frac{1.01^{n+1} - 1}{0.01} - n \right) \] which matches the given formula for part (a).

Calculate First Eight Terms of the Sequence for Part (b)

Using the formula from part (a), calculate interest for each quarter from 1 to 8: - For \( n=1 \): \( 10 \times \left( \frac{1.01^2-1}{0.01} - 1 \right) \)- For \( n=2 \): \( 10 \times \left( \frac{1.01^3-1}{0.01} - 2 \right) \)- ... - For \( n=8 \): \( 10 \times \left( \frac{1.01^9-1}{0.01} - 8 \right) \)Evaluate each expression to get numerical results.

Calculate Interest After 2 Years for Part (c)

After 2 years, or 8 quarters, use the formula: \[ Interest = 10 \left( \frac{1.01^9 - 1}{0.01} - 8 \right) \] Calculate the expression to find the interest accumulated after 8 quarters.

Key concepts

Annuity Future Value

The annuity future value is the amount of money accumulated after a series of periodic payments have been made, along with interest accrued on those payments. It's calculated using the formula:

• \( FV = P \times \frac{(1 + r)^n - 1}{r} \)

where \( P \) represents the regular deposit per period, \( r \) is the interest rate per period, and \( n \) is the number of periods. In this scenario, we deposit $10 at the beginning of each quarter.
This consistent deposit grows over time because of the interest given by the bank. The future value formula allows us to find the total accumulated amount, including both contributions and interest, by summing up these payments over \( n \) quarters.

Compounded Quarterly

Compounded quarterly means that interest is added to the deposit amounts four times a year. This regular addition of interest results in compound interest, where each new calculation of interest includes the previous interest as part of the principal.
In our example, the annual interest rate is 4%, so the quarterly interest rate, \( r \), is calculated as \( \frac{0.04}{4} = 0.01 \). Every quarter, the interest is added on, creating a snowball effect where future interest calculations are based on the new balance. This leads to exponential growth of the deposit amount over time.
Understanding the concept of compounding helps you see how deposits grow faster than with simple interest since interest is calculated more frequently.

Interest Accumulation

Interest accumulation refers to the growth of an initial sum deposited into an account over a period of time because of added interest. In the given exercise, interest is accumulated each quarter, enhancing the initial deposits.
To find the accumulated interest after \( n \) quarters, we subtract the total initial deposits from the future value:

• \[ 10 \times \left( \frac{(1.01)^n - 1}{0.01} - n \right) \]

The resulting formula captures just the interest part, excluding the initial money deposited. Over multiple quarters, this formula demonstrates how the interest on the interest grows, creating a larger balance than just your deposits.
Through this method, we determine how much additional money has been made purely from interest, which is crucial for long-term financial growth.

Sequence Terms

In this context, each term of the sequence represents the accumulated interest at the end of each quarter. The exercise requires us to calculate the first eight terms using the formula discussed in the solution.
Each term can be calculated through:

• For \( n=1 \): \( 10 \times \left( \frac{1.01^2-1}{0.01} - 1 \right) \)
• For \( n=2 \): \( 10 \times \left( \frac{1.01^3-1}{0.01} - 2 \right) \)
• ...\
• For \( n=8 \): \( 10 \times \left( \frac{1.01^9-1}{0.01} - 8 \right) \)

Calculating these terms shows how much interest has been accumulated by the end of each period. By tracking these terms, you can identify trends in interest growth and gain insights into the effectiveness of investing money regularly. This sequence is a critical part of understanding how the regular deposits and compound interest build wealth over time.