Question

James begins a savings plan in which he deposits \(\$ 100\) at the beginning of each month into an account that earns \(9 \%\) interest annually or, equivalently, \(0.75 \%\) per month. To be clear, on the first day of each month, the bank adds \(0.75 \%\) of the current balance as interest, and then James deposits \(\$ 100\). Let \(B_{n}\) be the balance in the account after the \(n\) th deposit, where \(B_{0}=\$ 0\). a. Write the first five terms of the sequence \(\left\\{B_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{B_{n}\right\\}\). c. How many months are needed to reach a balance of \(\$ 5000 ?\)

Short answer

Question: Calculate the first five terms of the sequence \(\{B_n\}\), find a recurrence relation for it and determine the number of months needed to reach a balance of \(5000\). Answer: The first five terms of the sequence are approximately \(100.75, 201.51, 302.27, 403.04, 503.82\). The recurrence relation for the sequence is \(B_{n}= (1 + 0.0075)B_{n-1} + 100\) with the initial condition \(B_0 = 0\). It takes James 47 months to reach a balance of \(5000\).

Step-by-step solution

a. First five terms of the sequence \(\{B_n\}\)

To find the first five terms of the sequence \(\{B_n\}\), we can simply follow the problem statement. Given \(B_0 = 0\), we can calculate the next terms as follows: 1. \(B_1 = 0 + 0.0075(100) + 100 = 100.75\) 2. \(B_2 = 100.75 + 0.0075(100.75) + 100 = 201.50625\) 3. \(B_3 = 201.50625 + 0.0075(201.50625) + 100 = 302.2694375\) 4. \(B_4 = 302.2694375 + 0.0075(302.2694375) + 100 = 403.0399315625\) 5. \(B_5 = 403.0399315625 + 0.0075(403.0399315625) + 100 = 503.817694363\) So the first five terms of the sequence are approximately: \(100.75, 201.51, 302.27, 403.04, 503.82\).

b. A recurrence relation for the sequence \(\{B_n\}\)

To find a recurrence relation for the sequence \(\{B_n\}\), we need to find a formula for \(B_n\) in terms of previous balances. By analyzing the problem, we can derive the following formula: \(B_n = B_{n-1} + 0.0075(B_{n-1}) + 100\) This formula computes the balance at the nth deposit using the balance of the previous deposit, after adding the interest and the new deposit of \(100\). Therefore, the recurrence relation for the sequence is: \(B_{n}= (1 + 0.0075)B_{n-1} + 100\) with the initial condition \(B_0 = 0\).

c. Number of months needed to reach a balance of \(5000\)

To find how many months are needed to reach a balance of \(5000\), we can use the recurrence relation we found in part b: \(B_{n}= (1 + 0.0075)B_{n-1} + 100\) We will keep calculating the balance using this formula until we reach a balance of \(5000\). Let's estimate the number of months needed: Starting with \(B_0 = 0\), we keep calculating \(B_n\) until \(B_n >= 5000\): \(B_1 = 1.0075(0) + 100 \approx 100.75 \) \(B_2 = 1.0075(100.75) + 100 \approx 201.51 \) \(...\) By checking, we will find that the balance surpasses \(5000\) after the \(47\)th deposit: \(B_{47} \approx 5026.08\) So, it takes \(47\) months for James to reach a balance of \(5000\).

Key concepts

Understanding the Savings Plan

A savings plan is essentially a strategy to accumulate wealth over time. For someone like James, who decides to make regular deposits into an account, understanding how the savings plan contributes to the growth of his funds is crucial. In this scenario, James's plan involves depositing \(100 into his account at the start of every month at an account with a 0.75% monthly interest rate.

In the context of our problem, a savings plan like James's can be modeled using a sequence of balances (\(B_n\) where each term represents the account balance after each deposit. It's essential to recognize that this isn't just a simple addition of \)100 each month; the interest accrued on the current balance must also be factored in, which introduces the concept of compound interest into the equation.

Students can benefit from simulating this process step-by-step for the initial few months, as done in the solution, to get a clear picture of how the balance increases not linearly, but at an increasing rate due to interest accumulation. It's important to provide this practical example of the benefits of regular savings coupled with the power of compound interest—a fundamental concept in personal finance.

Decoding Interest Calculation

Interest calculation is at the heart of understanding how investments grow over time. To put it simply, interest is the cost of using someone else's money or the reward for saving and lending your own. In our exercise, we are dealing with a bank account that earns compound interest – a critical concept where interest is earned not only on the initial amount deposited but also on the accrued interest from previous periods.

Monthly compounding, as in James's account, means that each month's interest is added to the balance and will itself earn interest in the following months. The formula for calculating this in the context of James's savings can be represented as \( B_n = (1 + r)B_{n-1} + P \) where \( r \) is the monthly interest rate expressed as a decimal, \( B_{n-1} \) is the previous month's balance, and \( P \) is the monthly deposit amount. Using this formula allows one to forecast the future value of savings based on consistent deposits and a fixed interest rate, which is a valuable skill in financial planning.

Sequence and Series in Finance

A sequence, in mathematical terms, is an ordered list of numbers where each number is called a term. In a financial context like James's savings plan, the sequence represents the account balance at different points in time. Recurrence relations like \( B_{n}= (1 + r)B_{n-1} + P \) embody such sequences, providing a rule to go from one term to the next by considering the previous term and some consistent changes—in this case, adding interest and deposit.

A series, in contrast, is the sum of the terms of a sequence, which would represent the cumulative amount that James has effectively put into the account, including both deposits and interest. Understanding series can help students grasp the total value of periodic investments and the impact of regular contributions over time. Modeling savings plans using sequences and series provides a detailed insight into how individual deposits and compounding contribute to wealth accumulation—an enlightening perspective for anyone embarking on a journey of financial growth. Moreover, applying these concepts enables students to compute future balances and investment goals with precision.