{Use of Tech} A savings plan
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 Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \(0.


c.How many months are needed to reach a balance of \)5000?

Verified step by step guidance

1. Recognize that the balance after the nᵗʰ payment, \(B_n\), follows a recurrence relation because each month the bank first adds interest to the current balance, then James deposits \$100.
2. Express the recurrence relation as: \[B_n = B_{n-1} \times (1 + 0.0075) + 100,\] where \(0.0075\) is the monthly interest rate (0.75%).
3. Identify that this is a non-homogeneous linear recurrence relation, which can be solved using the formula for the sum of a geometric series or by recognizing it as a future value of an annuity with monthly compounding.
4. Write the explicit formula for \(B_n\) after \(n\) months: \[B_n = 100 \times \frac{(1 + 0.0075)^n - 1}{0.0075}.\] This formula comes from summing the compounded deposits over \(n\) months.
5. Set \(B_n = 5000\) and solve for \(n\): \[5000 = 100 \times \frac{(1 + 0.0075)^n - 1}{0.0075}.\] Rearrange to isolate \((1 + 0.0075)^n\) and then take the natural logarithm to solve for \(n\).

Key Concepts

Compound Interest with Periodic Deposits

Compound interest means interest is earned on both the initial principal and the accumulated interest from previous periods. When regular deposits are made, each deposit grows with interest over time, creating a series of compounded amounts. Understanding how interest is applied before or after deposits is crucial for accurate calculations.

Recurrence Relation for Account Balance

The balance after each month can be expressed using a recurrence relation: the new balance equals the previous balance plus interest, then plus the new deposit. This relation helps model the growth of the account step-by-step and is essential for finding the balance after n months or determining n for a target balance.

Solving for Number of Periods in Annuity Problems

To find how many months are needed to reach a target balance, one uses formulas for the future value of an annuity with compound interest. This involves solving for the number of periods n in the equation that relates periodic deposits, interest rate, and final balance, often requiring logarithms or iterative methods.