A person decided to invest money at the end of each quarter. His initial investment amount is R998 but thereafter, starting with the second quarter, he decreased the amount by R34 each time. The accumulated investment after 2 years if interest is compounded at i(4) = 18% p.a., is? 

This problem involves a decreasing arithmetic sequence of investments, compounded quarterly, with interest compounded at a nominal rate of \( i^{(4)} = 18\% \) per annum.

### Key variables:
- Initial investment (at the end of the first quarter): \( R_1 = 998 \) 
- Each subsequent investment decreases by \( 34 \) each quarter.
- Time period: 2 years (which is \( 8 \) quarters)
- Interest rate per quarter: \( i^{(4)} = \frac{18\%}{4} = 4.5\% = 0.045 \)

### Step-by-step approach:

1. **Define the sequence of investments**:
   The investments form an arithmetic sequence where:
   \[
   R_n = 998 - 34(n-1)
   \]
   for each quarter \( n \).

   So the first few investments are:
   - Quarter 1: \( 998 \)
   - Quarter 2: \( 998 - 34 = 964 \)
   - Quarter 3: \( 998 - 2 \times 34 = 930 \)
   - etc., until the 8th quarter.

2. **Future value of each investment**:
   Each investment accumulates interest over the remaining quarters until the end of the 2 years. The formula for the future value (FV) of an investment made at the end of quarter \( n \), for \( t \) quarters at interest rate \( i \), is:
   \[
   FV = R_n \times (1 + i)^t
   \]
   where \( t \) is the number of remaining quarters for each investment to grow.

3. **Calculating the future value for each investment**:
   We sum the future values of all the investments. Here's the breakdown:
   
   - For \( n = 1 \) (first investment), \( t = 8 - 1 = 7 \):
     \[
     FV_1 = 998 \times (1.045)^7
     \]
   - For \( n = 2 \) (second investment), \( t = 8 - 2 = 6 \):
     \[
     FV_2 = 964 \times (1.045)^6
     \]
   - For \( n = 3 \) (third investment), \( t = 8 - 3 = 5 \):
     \[
     FV_3 = 930 \times (1.045)^5
     \]
   - Continue this for all 8 quarters until the last one.

4. **Final answer**:
   By calculating and summing all the future values from quarter 1 to quarter 8, we obtain the total accumulated investment after 2 years.

Let me calculate the exact value for you:The total accumulated investment after 2 years, with quarterly investments that decrease by R34 each time and compounded at an interest rate of 18% per annum (4.5% per quarter), is approximately **R8318.58**.

Would you like further details or clarifications on the process? Here are five related questions to expand on this topic:

1. How would the accumulated value change if the interest rate were different?
2. What if the initial investment was higher or lower, how would that impact the outcome?
3. Can you calculate the total investment without interest for comparison?
4. How does the frequency of compounding affect the final amount?
5. What is the formula for compound interest, and how does it apply to real-life investments?

**Tip**: When dealing with compound interest, always check whether the compounding frequency matches the periods of your cash flow (monthly, quarterly, annually). It significantly affects the future value.

A person decided to invest money at the end of each quarter. His initial investment amount is R998 but thereafter, starting with the second quarter, he decreased the amount by R34 each time. The accumulated investment after 2 years if interest is compounded at i(4) = 18% p.a., is?

This problem involves calculating the future value of a decreasing quarterly investment with an initial amount of R998, where each subsequent investment decreases by R34. Interest is compounded at 18% per annum. Learn how to apply arithmetic sequences and compound interest formulas to solve this problem.

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