In saving for a future major​ purchace, Oksana has been making uniform deposits of ​$4 comma 000 per year in an account that has earned variable annual interest rates of i1​ = i2​ = i3​ = 5​% and i4​ = i5 ​= 7​% ​(see cash flow diagram​ below). How much does she have in the account immediately after making the last​ deposit?

Oksana has been making uniform deposits of $4,000 per year, with varying interest rates. To calculate the amount in her account after the last deposit, we need to handle the two distinct interest rate periods separately: for the first three years, the interest rate is 5%, and for the last two years, it is 7%.

### Step-by-step breakdown:

1. **For years 1-3 (5% interest rate)**:
   Each $4,000 deposit will accumulate interest over the remaining years, depending on when the deposit was made. We will calculate the future value (FV) of each deposit made during this period using the formula for compound interest:

   \[
   \text{Future Value (FV)} = P \times (1 + i)^n
   \]

   where:
   - $P = 4,000$ (the deposit amount),
   - $i = 0.05$ (the interest rate for the first 3 years),
   - $n$ is the number of years the deposit will compound.

   - The deposit made at the end of year 1 will accumulate for 4 years (i.e., 3 years at 5%, plus 2 years at 7%).
   - The deposit made at the end of year 2 will accumulate for 3 years (i.e., 2 years at 5%, plus 1 year at 7%).
   - The deposit made at the end of year 3 will accumulate for 2 years (i.e., 1 year at 5%, plus 1 year at 7%).

2. **For years 4-5 (7% interest rate)**:
   Each $4,000 deposit during these years will only earn interest at 7%, and we'll calculate the future value based on how long the deposit remains in the account.
   - The deposit made at the end of year 4 will accumulate for 1 year at 7%.
   - The deposit made at the end of year 5 will not accumulate any interest because it is made at the end of the period.

Now, let’s calculate the future value for each deposit.

### Calculations:

1. **First deposit (Year 1)**:
   \[
   \text{FV}_1 = 4,000 \times (1 + 0.05)^3 \times (1 + 0.07)^2
   \]

2. **Second deposit (Year 2)**:
   \[
   \text{FV}_2 = 4,000 \times (1 + 0.05)^2 \times (1 + 0.07)^1
   \]

3. **Third deposit (Year 3)**:
   \[
   \text{FV}_3 = 4,000 \times (1 + 0.05)^1 \times (1 + 0.07)^1
   \]

4. **Fourth deposit (Year 4)**:
   \[
   \text{FV}_4 = 4,000 \times (1 + 0.07)^1
   \]

5. **Fifth deposit (Year 5)**:
   \[
   \text{FV}_5 = 4,000
   \]

### Total Future Value:
The total amount in the account is the sum of the future values of all deposits:

\[
\text{Total Future Value} = \text{FV}_1 + \text{FV}_2 + \text{FV}_3 + \text{FV}_4 + \text{FV}_5
\]

I will now compute these values for you.The total amount in Oksana's account immediately after making the last deposit is **$22,794.16**.

Would you like further details or have any questions about the calculations?

Here are 5 related questions to further explore the topic:
1. How would the total amount change if the interest rate for the last two years was 8% instead of 7%?
2. What if Oksana made semi-annual deposits instead of annual ones? How would that impact the final amount?
3. How much more would Oksana have if the interest rate was constant at 7% for all five years?
4. What happens if she deposits $4,000 annually but skips a year? How would that affect the total?
5. How much would she have after five years if there was no interest at all?

**Tip:** The longer the time money is invested, the greater the impact of compound interest, especially with increasing interest rates.

In saving for a future major purchase, Oksana has been making uniform deposits of $4,000 per year in an account that has earned variable annual interest rates of i1 = i2 = i3 = 5% and i4 = i5 = 7%. How much does she have in the account immediately after making the last deposit?

Learn how to calculate the future value of $4,000 annual deposits with variable interest rates of 5% and 7%. This detailed breakdown covers compound interest and future value calculations for financial mathematics, suitable for Grades 10-12.

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