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 payment, 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. Determine how many months are needed to reach a balance of \(\$ 5000\).

Short answer

In this problem, James deposits $100 into a savings account each month, and the account has a 0.75% monthly interest rate. We calculated the first five terms of the account balance sequences, from which we found they are approximately {100, 200.75, 302.51, 405.28, 509.06}. We also determined a recurrence relation, which is \(B_{n} = B_{n-1} \times (1+0.0075) + \$ 100\). Lastly, we calculated that it takes about 52 months for James' account balance to reach or exceed $5000.

Step-by-step solution

Calculate the First Five Terms

We will create a table to help visualize the problem, and then calculate each account balance given the conditions: - Month 0: \(B_{0} = \$ 0\) - Month 1: \(B_{1} = B_{0} \times (1+0.0075) + \$ 100 =\$ 100\) - Month 2: \(B_{2} = B_{1} \times (1+0.0075) + \$ 100 = \$ 200.75\) - Month 3: \(B_{3} = B_{2} \times (1+0.0075) + \$ 100 = \$ 302.505625\) - Month 4: \(B_{4} = B_{3} \times (1+0.0075) + \$ 100 = \$ 405.276216875\) - Month 5: \(B_{5} = B_{4} \times (1+0.0075) + \$ 100 = \$ 509.06004915625\) So the first five terms of the sequence are: \(\{100, 200.75, 302.505625, 405.276216875, 509.06004915625\}\).

Find a Recurrence Relation

To find a recurrence relation that generates the sequence, we can use the pattern we observed in the previous step: \(B_{n} = B_{n-1} \times (1+0.0075) + \$ 100\) This is the recurrence relation for calculating the balance in James' account.

Determine the Number of Months Needed to Reach a Balance of \(5000\)

We will use the recurrence relation to find how many months it takes for the account balance to reach or exceed $5000: 1. Initialize the account balance \(B_{0} = 0\) and the counter \(n=0\). 2. Calculate the next balance using the recurrence relation \(B_{n} = B_{n-1} \times (1+0.0075) + \$ 100\). 3. Check if the balance has reached or exceeded \(5000\). If so, stop the process and report the result as \(n\) months. If not, increment the counter by 1 (\(n = n+1\)) and go back to step 2. Using this method, we find that James' account balance will exceed \(5000 after approximately \)52$ months.

Key concepts

Savings Plan

Engaging in a savings plan is a proactive step towards securing your financial future. In this case, James has opted for a sensible strategy by depositing a fixed amount of \(\\(100\) at the beginning of every month into a savings account.
This consistent deposit method helps increase savings progressively and takes advantage of compound interest.

• Regular deposits help in formulating a habit of saving and can simplify financial management over time.
• Starting with **B**\(_{0} = \\)0\)**, we see how small, consistent deposits can grow into substantial sums.

This method of savings encourages individuals to look forward to the gradual increase of their funds and exploit the benefits of interest earnings. This leads seamlessly into the concept of interest calculation.

Interest Calculation

Interest calculation is pivotal in understanding how savings can grow over time. Here, James benefits from an annual interest rate of \(9\%\), which translates to a monthly rate of \(0.75\%\), as banks typically compound interest on a regular basis.
Understanding this compounding mechanism is essential.

• The formula for each month's balance includes the previous balance, interest, and the new deposit.
• Calculate the interest as a percentage of the current balance, adding it before making the month's deposit. For instance, from Month 1 to 2: \[B_{2} = B_{1} \times (1+0.0075) + \\(100 = \\)200.75\]

Recognizing how interest compounds can vastly improve financial literacy and decision-making, ensuring that money earns income over time without any active effort from the account holder. With this iterative process, we move towards generating sequences in recurrence relations.

Sequence Generation

A sequence refers to a set of numbers arranged in a specific order based on a discernible pattern. In James’ savings plan, the sequence is generated using a recurrence relation.
This relation helps in forecasting the future balance by applying a consistent formula repeatedly:

• The sequence \(\left\{B_{n}\right\}\) is derived using the recurrence relation \[B_{n} = B_{n-1} \times (1+0.0075) + \\(100\]
• This deepens understanding of how sequences can model real-world scenarios like savings growth.

The practical application of sequence generation here allows us to predict that reaching a balance of \(\\)5000\) would take approximately 52 months. This prediction not only boosts confidence in our calculations but also in planning financial goals effectively.