The following sequence is given below:
`1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,….` etc.
The sum of the first 100 terms of this sequence is :

Options

A. a. 945
B. b. 358350
C. c. 100!
D. d. none of these

Answer

To find the sum of the first 100 terms of the given sequence, we can break down the sequence and analyze how many times each number appears.

### Step-by-step Solution:

1. **Understanding the Sequence:**
The sequence is structured such that:
- The number 1 appears 1 time.
- The number 2 appears 2 times.
- The number 3 appears 3 times.
- The number 4 appears 4 times.
- The number 5 appears 5 times.
- The number 6 appears 6 times.
- And so on...

2. **Finding the Total Number of Terms:**
To determine how far we need to go in the sequence to reach 100 terms, we can calculate the cumulative total of terms:
- For 1: 1 term (Total = 1)
- For 2: 2 terms (Total = 1 + 2 = 3)
- For 3: 3 terms (Total = 3 + 3 = 6)
- For 4: 4 terms (Total = 6 + 4 = 10)
- For 5: 5 terms (Total = 10 + 5 = 15)
- For 6: 6 terms (Total = 15 + 6 = 21)
- For 7: 7 terms (Total = 21 + 7 = 28)
- For 8: 8 terms (Total = 28 + 8 = 36)
- For 9: 9 terms (Total = 36 + 9 = 45)
- For 10: 10 terms (Total = 45 + 10 = 55)
- For 11: 11 terms (Total = 55 + 11 = 66)
- For 12: 12 terms (Total = 66 + 12 = 78)
- For 13: 13 terms (Total = 78 + 13 = 91)
- For 14: 14 terms (Total = 91 + 14 = 105)

We see that up to 13, we have 91 terms. The next number, 14, will take us to 105 terms, which exceeds 100.

3. **Calculating the Additional Terms Needed:**
Since we have 91 terms from numbers 1 to 13, we need 9 more terms to reach 100. Thus, we will take the number 14 for 9 times.

4. **Calculating the Sum of the First 100 Terms:**
Now we calculate the sum of the first 100 terms:
- The sum of numbers from 1 to 13:
\[ \text{Sum} = 1 + 2 + 3 + ... + 13 = \frac{n(n + 1)}{2} = \frac{13(13 + 1)}{2} = \frac{13 \times 14}{2} = 91 \]
- The contribution from the number 14 (which appears 9 times):
\[ \text{Contribution from 14} = 14 \times 9 = 126 \]
- Therefore, the total sum of the first 100 terms is:
\[ \text{Total Sum} = 91 + 126 = 217 \]

### Final Answer:
The sum of the first 100 terms of the sequence is **217**.