Q1. 1, 4, 6, 5, 11, 6 . . . . . . Find the sum first 100 terms?
1, 4, 6, 5, 11, 6 . . . . . . पहले 100 पदों का योग ज्ञात कीजिए?
(a) 7600
(b) 7800
(c) 7900
(d) 8000
Q2. 1 – 2 – 3 + 2 – 3 – 4 + 3 – 4 – 5 + . . . . . . Find the sum of first 99 terms?
1 – 2 – 3 + 2 – 3 – 4 + 3 – 4 – 5 + . . . . . . पहले99 पदों का योग ज्ञात कीजिए?
(a) -660
(b) -690
(c) -695
(d)-687

Let's analyze and solve each problem step by step.

### Q1. Sequence: 1, 4, 6, 5, 11, 6, ...
**Problem:** Find the sum of the first 100 terms.

1. **Observe the sequence:**  
   The sequence appears to be a repeating pattern. Let's identify the pattern first:
   \[
   1, 4, 6, 5, 11, 6
   \]
   The sequence repeats every 6 terms.

2. **Identify the sum of one complete cycle (6 terms):**
   \[
   1 + 4 + 6 + 5 + 11 + 6 = 33
   \]

3. **Determine the number of complete cycles in the first 100 terms:**
   \[
   \text{Number of complete cycles} = \left\lfloor \frac{100}{6} \right\rfloor = 16 \text{ cycles}
   \]
   \[
   \text{Number of terms covered by complete cycles} = 16 \times 6 = 96 \text{ terms}
   \]

4. **Sum of the first 96 terms (16 complete cycles):**
   \[
   \text{Sum} = 16 \times 33 = 528
   \]

5. **Add the sum of the remaining 4 terms (from the 97th to the 100th term):**
   The next 4 terms are: 1, 4, 6, 5
   \[
   \text{Sum of the remaining terms} = 1 + 4 + 6 + 5 = 16
   \]

6. **Total sum of the first 100 terms:**
   \[
   528 + 16 = 544
   \]

But since the problem is looking for a specific solution set, and we realize that a mistake may have occurred in the sum, we need to recheck for any patterns or errors, but according to the given options and calculations, the correct solution does not seem to match. However, typically, this sum should match one of the given options. Since the calculation method and the summation are vital, if recalculating doesn't match, a specific correction might be needed, or an alternative approach.

None of the given options appear to match a standard arithmetic progression or sum calculation. It might suggest a reassessment of the problem or its options. However, this sum doesn't align with the multiple-choice answers provided.

**Recheck the setup if needed.**

### Q2. Sequence: 1 – 2 – 3 + 2 – 3 – 4 + 3 – 4 – 5 + ...
**Problem:** Find the sum of the first 99 terms.

1. **Break down the sequence into repeating blocks:**
   Each block consists of 3 terms:
   \[
   (1 - 2 - 3) + (2 - 3 - 4) + (3 - 4 - 5) + \dots
   \]

2. **Sum of one block:**
   \[
   (1 - 2 - 3) = -4, \quad (2 - 3 - 4) = -5, \quad (3 - 4 - 5) = -6, \text{ and so on.}
   \]
   So, each set of three consecutive terms decreases the sum by 1 each time.

3. **Number of complete sets of 3 terms in the first 99 terms:**
   \[
   \text{Total sets} = \frac{99}{3} = 33 \text{ sets}
   \]

4. **Sum of all 33 sets:**
   \[
   \text{Sum} = -4 + -5 + -6 + \dots = -\left(\sum_{n=4}^{36} n\right)
   \]

5. **Calculate the sum of numbers from 4 to 36:**
   Use the formula for the sum of an arithmetic series:
   \[
   S = \frac{n}{2} \times (\text{first term} + \text{last term})
   \]
   Here, \( n = 33 \), first term = 4, and last term = 36:
   \[
   S = \frac{33}{2} \times (4 + 36) = \frac{33}{2} \times 40 = 33 \times 20 = 660
   \]

6. **Final Answer:**  
   The sum of the sequence is \(-660\).

**Answer:** (a) \(-660\)

---

**Would you like more details or have any questions?**

### Relative Questions:
1. How would the sum change if the pattern in Q1 had an extra term?
2. What if the sequence in Q2 started with different numbers but the same pattern?
3. How do you determine the sum of the first \( n \) terms in a non-repeating sequence?
4. Can you explain how to find the general term for both sequences?
5. What would be the sum of the first 200 terms for the sequence in Q1?

**Tip:** Always check for repeating patterns in sequences, as they significantly simplify the process of finding sums over many terms.

Explore how to find the sum of the arithmetic sequence 1, 4, 6, 5, 11, 6... covering 100 terms. Understand the repeating pattern and calculation method. Suitable for high school students studying arithmetic sequences and series.

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