Zahrul also gave money to the charity over the same 20-year period. He gave RMA in Year
1 and the amounts of money he gave each year increased with common difference RMB.
The total amount of money that Zahrul gave over the 20-year period was less RM900 from
the total amount that Natasha gave,RM4900. The amount of money Zahrul gave in Year 10 was
RM190.
(c) Calculate the values of A and B.

(d) In a 30-year period, who gave more money to the charity.

Given the problem details, let's proceed to calculate the values of \( A \) and \( B \) and then determine who gave more money over a 30-year period.

### (c) Calculate the values of \( A \) and \( B \)

#### Step 1: Information Recap

- Zahrul's contributions follow an arithmetic sequence:
  - The first term \( a_1 = A \).
  - The common difference \( d = B \).
- The amount given in Year 10 is \( a_{10} = 190 \).
- The total amount given over 20 years by Zahrul was RM900 less than Natasha's total, \( S_{20} = 4900 - 900 = 4000 \).

#### Step 2: Use the Formula for the 10th Term

For an arithmetic sequence, the \( n \)-th term is given by:
\[
a_n = a_1 + (n-1)d
\]
For Year 10:
\[
a_{10} = A + 9B
\]
Given \( a_{10} = 190 \):
\[
A + 9B = 190 \quad \text{(Equation 1)}
\]

#### Step 3: Use the Formula for the Sum of 20 Years

The sum of the first \( n \) terms of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} \times [2a_1 + (n-1)d]
\]
For 20 years:
\[
S_{20} = \frac{20}{2} \times [2A + 19B] = 10 \times (2A + 19B)
\]
Given that \( S_{20} = 4000 \):
\[
10(2A + 19B) = 4000
\]
\[
2A + 19B = 400 \quad \text{(Equation 2)}
\]

#### Step 4: Solve the System of Equations

Now, solve the system of linear equations:

1. \( A + 9B = 190 \)
2. \( 2A + 19B = 400 \)

**From Equation 1:**

\[
A = 190 - 9B
\]

**Substitute into Equation 2:**

\[
2(190 - 9B) + 19B = 400
\]
\[
380 - 18B + 19B = 400
\]
\[
B = 400 - 380 = 20
\]

Now, substitute \( B = 20 \) back into the equation for \( A \):

\[
A = 190 - 9(20) = 190 - 180 = 10
\]

So, the values are:
\[
A = 10, \quad B = 20
\]

### (d) In a 30-Year Period, Who Gave More Money to the Charity?

#### Step 1: Calculate Zahrul's Contribution Over 30 Years

Using the same formula for the sum of an arithmetic sequence:
\[
S_{30} = \frac{30}{2} \times [2A + 29B] = 15 \times (2A + 29B)
\]
Substitute the values of \( A \) and \( B \):
\[
S_{30} = 15 \times [2(10) + 29(20)]
\]
\[
S_{30} = 15 \times [20 + 580] = 15 \times 600 = 9000
\]

#### Step 2: Natasha's Contribution Over 30 Years

Since Natasha's total contribution over 20 years was RM4900, assuming the same pattern for the next 10 years without additional information (assuming her contributions continue at the same rate as the previous 10 years), Zahrul’s RM9000 is higher than Natasha’s RM4900.

Thus, **Zahrul gave more money over the 30-year period**.

---

### 5 Related Questions:
1. What is the general form of an arithmetic sequence?
2. How does the common difference in an arithmetic sequence affect the total sum?
3. What would happen if the common difference \( B \) was negative?
4. How can you determine the total amount given over 40 years?
5. How does an arithmetic sequence compare to a geometric sequence in terms of growth?

### Tip:
When working with sequences, always verify each term and sum formula against known values to ensure accuracy, especially for longer periods like 30 years.

Explore how to calculate Zahrul's contributions to charity over a 30-year period with detailed step-by-step solutions. This problem involves arithmetic sequences and series summation, suitable for high school students.

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