In the sequence 1, 2, 2, 3, 3, 3, 4, 4,4,4,....., where n consecutive terms have the value n, the 150 term is

Options

A. 17
B. 16
C. 18
D. none of these

Explanation

To find the 150th term in the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ..., where each number n appears n times, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Sequence**:
- The sequence starts with 1 appearing once, 2 appearing twice, 3 appearing three times, and so on.
- The pattern continues such that the number n appears n times.

2. **Finding the Position of the 150th Term**:
- We need to determine the value of n such that the total number of terms up to n is at least 150.
- The total number of terms from 1 to n can be calculated using the formula for the sum of the first n natural numbers:
\[ S_n = \frac{n(n + 1)}{2} \]
- We want to find n such that:
\[ S_n \geq 150 \]

3. **Setting Up the Inequality**:
- We set up the inequality:
\[ \frac{n(n + 1)}{2} \geq 150 \]
- Multiplying both sides by 2 gives:
\[ n(n + 1) \geq 300 \]

4. **Finding the Roots**:
- We can rearrange this to find the roots:
\[ n^2 + n - 300 \geq 0 \]
- To find the roots of the quadratic equation \( n^2 + n - 300 = 0 \), we can use the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 1, b = 1, c = -300 \):
\[ n = \frac{-1 \pm \sqrt{1 + 1200}}{2} = \frac{-1 \pm \sqrt{1201}}{2} \]

5. **Calculating the Roots**:
- Approximating \( \sqrt{1201} \approx 34.64 \):
\[ n = \frac{-1 + 34.64}{2} \approx 16.82 \quad \text{(taking the positive root)} \]
- The negative root is not relevant since n must be positive.

6. **Determining the Integer Value**:
- Since n must be a whole number, we take \( n = 17 \) as the largest integer satisfying the inequality.

7. **Verifying the Total Number of Terms**:
- Calculate the total number of terms for \( n = 17 \):
\[ S_{17} = \frac{17 \times 18}{2} = 153 \]
- Calculate for \( n = 16 \):
\[ S_{16} = \frac{16 \times 17}{2} = 136 \]
- Since \( S_{16} < 150 < S_{17} \), the 150th term is indeed the number 17.

### Conclusion:
The 150th term in the sequence is **17**.