We are given that Alan deposits to an account at the end of every year for 12 years, with the first deposit being $300 and each subsequent deposit increasing by $50. The account earns an annual effective interest rate of 4%, and we need to find the accumulated value of the account after 12 years.

Step 1: Understanding the Deposit Sequence

The deposit sequence forms an arithmetic progression (AP) where:

• First deposit ($a$) = $300
• Common difference ($d$) = $50
• Number of deposits ($n$) = 12 years

So the deposits over the years are:

• $300, 350, 400, \ldots$ up to the 12th deposit.

The amount deposited in the 12th year is: $a_{12} = a + (n - 1) d = 300 + (12 - 1) \times 50 = 300 + 550 = 850$

Thus, the deposits form the sequence: $300, 350, 400, \ldots, 850$

Step 2: Formula for the Accumulated Value

The accumulated value after 12 years is the sum of the future values of all the deposits, where each deposit earns interest until the end of the 12th year.

The formula for the future value of each deposit is: $\text{Future Value} = \text{Deposit} \times (1 + i)^{(12 - k)}$ where $i = 0.04$ (the interest rate) and $k$ is the year of the deposit.

Step 3: Calculate the Accumulated Value

We can break this calculation into individual steps, summing the future values of each deposit. Let's calculate it step-by-step.

The accumulated value of Alan's account after 12 years, considering the increasing deposits and the 4% annual effective interest rate, is approximately $8,290.

Would you like a detailed breakdown of the steps involved, or do you have any further questions?

Here are five related questions you might find interesting:

1. How would the accumulated value change if the interest rate were 5% instead of 4%?
2. What is the total amount Alan deposited over the 12 years?
3. How would the result differ if the deposits were made at the beginning of each year instead of the end?
4. What is the accumulated value if the deposits increase by $100 instead of $50?
5. How would the accumulated value be affected if Alan skipped deposits in some years?

Tip: The formula for the future value of a series of payments is sensitive to the timing of deposits (beginning vs. end of period).