N, the set of natural numbers, is partitioned into subsets S1 ​ ={1},S2 ​ ={2,3},S3 ​ ={4,5,6},S4 ​ ={7,8,9,10}. The last term of these groups is 1,1+2,1+2+3,1+2+3+4, so on. Find the sum of the elements in the subset S50 ​ .

Options

A. a. `S_12`
B. b. `S_13`
C. c. `S_12 and S_13`
D. d. `S_9, S_12 ,S_(13)`

Answer

To solve the problem of finding the sum of the elements in the subset \( S_{50} \), we can follow these steps:

### Step 1: Identify the last term of each subset
The last term of each subset \( S_n \) can be expressed as the sum of the first \( n \) natural numbers. This is given by the formula:
\[ \text{Last term of } S_n = \frac{n(n + 1)}{2} \]

### Step 2: Determine the last term of \( S_{50} \)
Using the formula from Step 1, we can find the last term of \( S_{50} \):
\[ \text{Last term of } S_{50} = \frac{50(50 + 1)}{2} = \frac{50 \times 51}{2} = \frac{2550}{2} = 1275 \]

### Step 3: Determine the first term of \( S_{50} \)
To find the first term of \( S_{50} \), we need to find the last term of \( S_{49} \):
\[ \text{Last term of } S_{49} = \frac{49(49 + 1)}{2} = \frac{49 \times 50}{2} = \frac{2450}{2} = 1225 \]
Thus, the first term of \( S_{50} \) is \( 1226 \).

### Step 4: Identify the elements in \( S_{50} \)
The elements in \( S_{50} \) are the natural numbers starting from \( 1226 \) to \( 1275 \). The number of elements in \( S_{50} \) can be calculated as:
\[ \text{Number of elements} = 1275 - 1226 + 1 = 50 \]

### Step 5: Calculate the sum of the elements in \( S_{50} \)
The sum of the elements in \( S_{50} \) can be calculated using the formula for the sum of an arithmetic series:
\[ \text{Sum} = \frac{\text{Number of terms}}{2} \times (\text{First term} + \text{Last term}) \]
Substituting the values we found:
\[ \text{Sum} = \frac{50}{2} \times (1226 + 1275) = 25 \times 2501 = 62525 \]

### Final Answer
The sum of the elements in the subset \( S_{50} \) is \( 62525 \).

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