To calculate the accumulated value of the deposits at the end of 9 years with varying interest rates, we can use the following steps. Each deposit grows at the corresponding effective interest rate from the year it is made until the end of year 9.

Information:

• Interest rates:
  • 4% for years 1-3
  • 5% for years 4-6
  • 6% for years 7-9
• Deposits of $1,000 are made at the end of each year for 9 years.

Methodology:

Each deposit will grow from the year it is made to year 9. The accumulation factors are based on the interest rate applied for the years after each deposit:

1. Deposit in year 1:
   This deposit grows from the end of year 1 to the end of year 9.

   • Growth from year 1 to 3: $(1.04)^3$
   • Growth from year 4 to 6: $(1.05)^3$
   • Growth from year 7 to 9: $(1.06)^3$
     Total factor: $(1.04)^3 \times (1.05)^3 \times (1.06)^3$

2. Deposit in year 2:
   This deposit grows from the end of year 2 to the end of year 9.

   • Growth from year 2 to 3: $(1.04)^2$
   • Growth from year 4 to 6: $(1.05)^3$
   • Growth from year 7 to 9: $(1.06)^3$
     Total factor: $(1.04)^2 \times (1.05)^3 \times (1.06)^3$

3. Deposit in year 3:
   This deposit grows from the end of year 3 to the end of year 9.

   • Growth from year 3 to 3: $(1.04)^1$
   • Growth from year 4 to 6: $(1.05)^3$
   • Growth from year 7 to 9: $(1.06)^3$
     Total factor: $(1.04)^1 \times (1.05)^3 \times (1.06)^3$

4. Deposit in year 4:
   This deposit grows from the end of year 4 to the end of year 9.

   • Growth from year 4 to 6: $(1.05)^3$
   • Growth from year 7 to 9: $(1.06)^3$
     Total factor: $(1.05)^3 \times (1.06)^3$

5. Deposit in year 5:
   This deposit grows from the end of year 5 to the end of year 9.

   • Growth from year 5 to 6: $(1.05)^2$
   • Growth from year 7 to 9: $(1.06)^3$
     Total factor: $(1.05)^2 \times (1.06)^3$

6. Deposit in year 6:
   This deposit grows from the end of year 6 to the end of year 9.

   • Growth from year 6 to 6: $(1.05)^1$
   • Growth from year 7 to 9: $(1.06)^3$
     Total factor: $(1.05)^1 \times (1.06)^3$

7. Deposit in year 7:
   This deposit grows from the end of year 7 to the end of year 9.

   • Growth from year 7 to 9: $(1.06)^3$
     Total factor: $(1.06)^3$

8. Deposit in year 8:
   This deposit grows from the end of year 8 to the end of year 9.

   • Growth from year 8 to 9: $(1.06)^2$
     Total factor: $(1.06)^2$

9. Deposit in year 9:
   This deposit grows from the end of year 9 to the end of year 9.

   • Growth for year 9: $(1.06)^1$

Now, let me calculate the accumulated value based on these factors.The accumulated value of the deposits at the end of 9 years is approximately $11,793.09.

Would you like more details on any part of the solution?

Here are some related questions for further exploration:

1. What would the accumulated value be if the interest rates were compounded monthly?
2. How would the result change if a deposit was missed in year 5?
3. What if a lump-sum deposit of $10,000 was made at the beginning of year 1 instead of yearly deposits?
4. How would continuous compounding affect the accumulated value for the same interest rates?
5. What would be the effect of inflation on the real value of the account balance after 9 years?

Tip: When dealing with different interest rates over time, it's important to separate the time periods and apply the correct growth factors step by step to avoid errors.