This is a compound interest problem with additional regular deposits, also known as an annuity with compound interest. We'll break down the solution into steps:

Variables and Known Information:

1. Initial deposit $P_0 = 5000$ (January 1, 2008).
2. Monthly deposits of $D = 350$ for 4 years, so $12 \times 4 = 48$ months of deposits.
3. Final total amount $A = 22500$.
4. Compounding weekly, which means $n = 52$ times per year.
5. We need to find the annual interest rate $r$, which is compounded weekly.

Step 1: Calculate the Time Period

• The total period from January 1, 2008, to the end of December 2011 is 4 years.

Step 2: Formula for Compound Interest with Regular Deposits

The formula for compound interest with regular deposits is given by:

$A = P_0 \left(1 + \frac{r}{n}\right)^{nt} + D \left(\frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}\right)$

Where:

• $A$ is the final amount ($22500).
• $P_0$ is the initial principal ($5000).
• $r$ is the annual interest rate (unknown, what we need to find).
• $n$ is the number of compounding periods per year (52 for weekly compounding).
• $t$ is the total time in years (4 years).
• $D$ is the regular monthly deposit ($350, but we'll convert to weekly).

Step 3: Convert Monthly Deposit to Weekly

Since there are 52 weeks in a year, we need to convert the monthly deposit to a weekly equivalent.

• Monthly deposit $D = 350$.
• Weeks per month (approx) = $\frac{52}{12} \approx 4.33$.

The equivalent weekly deposit would be:

$D_w = \frac{350}{4.33} \approx 80.83$

Now we can apply this deposit in the weekly compounding formula.

Step 4: Solve for $r$

We will solve for the interest rate $r$ by substituting the known values into the formula. Since solving this equation directly involves complex algebra, we can use numerical methods to find the interest rate.

Let me compute that.The annual interest rate that Jim would need, compounded weekly, to reach a total of $22,500 after 4 years is approximately 1.28%.

Would you like any further clarification or details?

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