.A woman invests $350 at 6% simple interest at the beginning of each year for a period of 10 years. Find the total accumulated value of all the investments at the end of the 10-year period.

To solve this problem, we can calculate the total accumulated value of the investments at the end of the 10-year period using the formula for simple interest and then summing up the contributions.

### Step 1: Understanding Simple Interest
The formula for simple interest is given by:
\[
I = P \times r \times t
\]
Where:
- $ I $ is the interest earned,
- $ P $ is the principal amount,
- $ r $ is the annual interest rate (expressed as a decimal),
- $ t $ is the time in years.

### Step 2: Calculate the Accumulated Value for Each Investment
The woman invests $350 at the beginning of each year for 10 years. Each of these investments will earn interest for a different number of years, depending on when it was made.

Let's calculate the accumulated value for each year:

1. **1st year investment**: $ t = 10 $ years
   \[
   A_1 = 350 + (350 \times 0.06 \times 10) = 350 + 210 = 560
   \]
2. **2nd year investment**: $ t = 9 $ years
   \[
   A_2 = 350 + (350 \times 0.06 \times 9) = 350 + 189 = 539
   \]
3. **3rd year investment**: $ t = 8 $ years
   \[
   A_3 = 350 + (350 \times 0.06 \times 8) = 350 + 168 = 518
   \]
4. **4th year investment**: $ t = 7 $ years
   \[
   A_4 = 350 + (350 \times 0.06 \times 7) = 350 + 147 = 497
   \]
5. **5th year investment**: $ t = 6 $ years
   \[
   A_5 = 350 + (350 \times 0.06 \times 6) = 350 + 126 = 476
   \]
6. **6th year investment**: $ t = 5 $ years
   \[
   A_6 = 350 + (350 \times 0.06 \times 5) = 350 + 105 = 455
   \]
7. **7th year investment**: $ t = 4 $ years
   \[
   A_7 = 350 + (350 \times 0.06 \times 4) = 350 + 84 = 434
   \]
8. **8th year investment**: $ t = 3 $ years
   \[
   A_8 = 350 + (350 \times 0.06 \times 3) = 350 + 63 = 413
   \]
9. **9th year investment**: $ t = 2 $ years
   \[
   A_9 = 350 + (350 \times 0.06 \times 2) = 350 + 42 = 392
   \]
10. **10th year investment**: $ t = 1 $ year
    \[
    A_{10} = 350 + (350 \times 0.06 \times 1) = 350 + 21 = 371
    \]

### Step 3: Calculate the Total Accumulated Value
To find the total accumulated value, sum up the accumulated values for all the years:
\[
\text{Total Accumulated Value} = A_1 + A_2 + \dots + A_{10} = 560 + 539 + 518 + 497 + 476 + 455 + 434 + 413 + 392 + 371 = 4655
\]

### Final Answer
The total accumulated value of all the investments at the end of the 10-year period is **$4,655**.

Would you like any details or have any questions? 

Here are 5 related questions you might be interested in:

1. What is the formula for compound interest, and how does it differ from simple interest?
2. How would the total amount change if the interest were compounded annually?
3. What would be the total amount if the interest rate were increased to 8%?
4. How does the duration of investment affect the total accumulated value?
5. Can you calculate the future value of an annuity if the investments were made monthly instead of yearly?

**Tip:** In simple interest problems, the interest amount remains the same each year because it is calculated only on the original principal, not on any accumulated interest.

Learn how to calculate the total accumulated value of investments made at the beginning of each year with simple interest. This problem involves investing $350 annually at 6% interest for 10 years, covering financial mathematics concepts. Suitable for students and individuals interested in financial calculations.

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