You are running a foundation for street children. Mr. Orleans, a rich benefactor, offers two options for his donations:

Option A: To give ₽ 1,000.00 on day 1, ₽ 999 on day 2, ₽ 998 on day 3, with the process to end after 1,000 days.

Option B: To give ₽ 1.00 on day 1, ₽ 2.00 on day 2, ₽ 3.00 on day 3, for 19 days.

You have to tell him today which option you want to take.

To decide which option is better, let's calculate the total donation amount for both options.

### **Option A:**
Mr. Orleans offers ₽1,000 on the first day, then ₽999 on the second day, continuing in this way until the 1,000th day.

This series is an arithmetic sequence where:
- First term (a₁) = ₽1,000
- Common difference (d) = -₁ (since the donation decreases by ₽1 each day)
- Number of terms (n) = 1,000

The sum \( S_n \) of the first n terms of an arithmetic sequence can be calculated using the formula:

\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]

where \( a_n \) is the last term. In this case:
- \( a_n = a_1 + (n - 1) \times d = 1,000 + (1,000 - 1) \times (-1) = 1,000 - 999 = 1 \)

So the sum is:

\[
S_{1000} = \frac{1000}{2} \times (1,000 + 1) = 500 \times 1,001 = ₽ 500,500
\]

### **Option B:**
Mr. Orleans offers ₽1 on the first day, ₽2 on the second day, continuing this way for 19 days.

This series is another arithmetic sequence where:
- First term (a₁) = ₽1
- Common difference (d) = ₽1
- Number of terms (n) = 19

The sum of this sequence is:

\[
S_{19} = \frac{19}{2} \times (1 + 19) = \frac{19}{2} \times 20 = 19 \times 10 = ₽ 190
\]

### **Comparison:**
- **Option A** provides a total donation of ₽500,500.
- **Option B** provides a total donation of ₽190.

Clearly, **Option A** is the better choice, as it results in a much larger total donation.

Would you like to know more details or have any questions about this calculation?

Here are 8 questions for further exploration:
1. How do arithmetic sequences differ from geometric sequences?
2. What would be the sum if Option A continued for 2,000 days instead?
3. How does the formula for the sum of an arithmetic sequence work?
4. Can we calculate the total donation for a different time period in Option B?
5. What is the general approach to comparing financial options using sequences?
6. How does the rate of decrease in Option A affect the overall sum?
7. What would happen if Option B's terms doubled each day instead of increasing linearly?
8. What is the impact of choosing different values for the first term and common difference in sequences?

**Tip:** Always double-check the last term in an arithmetic sequence when using the sum formula, as it's key to accurate calculations.

Explore and compare two donation options using arithmetic sequences. Option A offers decreasing donations over 1,000 days, while Option B increases donations linearly over 19 days. Analyze the total donation amounts using arithmetic sequence formulas. Learn why Option A is the better choice for maximizing donation impact.

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