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 1st, 2nd , 4th , 7th , 11th , 16th , ..211 th` term is :

Options

A. 221
B. 400
C. 231
D. 211

Answer

To solve the problem of finding the sum of the 1st, 2nd, 4th, 7th, 11th, 16th, and 211th terms of the given sequence, we will follow these steps:

### Step 1: Understand the Sequence
The sequence is constructed such that the number \( n \) appears \( n \) times. For example:
- 1 appears 1 time: \( 1 \)
- 2 appears 2 times: \( 2, 2 \)
- 3 appears 3 times: \( 3, 3, 3 \)
- 4 appears 4 times: \( 4, 4, 4, 4 \)
- 5 appears 5 times: \( 5, 5, 5, 5, 5 \)
- And so on...

### Step 2: Identify the Terms
We need to find the specific terms in the sequence:
- 1st term: \( 1 \)
- 2nd term: \( 2 \)
- 4th term: \( 3 \)
- 7th term: \( 4 \)
- 11th term: \( 5 \)
- 16th term: \( 6 \)
- 211th term: To find this, we need to determine how many total terms there are up to each \( n \).

### Step 3: Calculate the Total Number of Terms
The total number of terms up to \( n \) can be calculated using the formula for the sum of the first \( n \) natural numbers:
\[ \text{Total terms} = \frac{n(n + 1)}{2} \]
We need to find \( n \) such that:
\[ \frac{n(n + 1)}{2} \geq 211 \]
Solving for \( n \):
\[ n(n + 1) \geq 422 \]
Testing values:
- For \( n = 20 \): \( 20 \times 21 / 2 = 210 \) (not enough)
- For \( n = 21 \): \( 21 \times 22 / 2 = 231 \) (sufficient)

Thus, the 211th term corresponds to \( 21 \) since \( 21 \) appears \( 21 \) times.

### Step 4: List the Terms
Now we have identified the terms:
- 1st term: \( 1 \)
- 2nd term: \( 2 \)
- 4th term: \( 3 \)
- 7th term: \( 4 \)
- 11th term: \( 5 \)
- 16th term: \( 6 \)
- 211th term: \( 21 \)

### Step 5: Calculate the Sum
Now we sum these identified terms:
\[ 1 + 2 + 3 + 4 + 5 + 6 + 21 \]
Calculating:
\[ 1 + 2 = 3 \]
\[ 3 + 3 = 6 \]
\[ 6 + 4 = 10 \]
\[ 10 + 5 = 15 \]
\[ 15 + 6 = 21 \]
\[ 21 + 21 = 42 \]

### Final Answer
The sum of the 1st, 2nd, 4th, 7th, 11th, 16th, and 211th terms is \( 42 \).

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