If `s_1 = (1), s_2 = (2) (3)`
`s_(3) = (4), (5), (6)`
`s_4 = (7), (8), (9), (10)`,
`s_5 = (11),(12),(13),(14),(15)`….etc.
Where `s_1 , s_2, s_3…..`, etc arc the first, second adn third terms of the given given sequence.
Then the sum of the numbers in `S_(50)`

Options

A. 62525
B. 57567
C. 62505
D. none of these

Answer

To solve the problem, we need to find the sum of the numbers in the set \( S_{50} \) based on the given sequence. Let's break it down step by step.

### Step 1: Identify the pattern in the sequence
The sets \( S_n \) contain increasing numbers of consecutive integers:
- \( S_1 = (1) \) has 1 number.
- \( S_2 = (2, 3) \) has 2 numbers.
- \( S_3 = (4, 5, 6) \) has 3 numbers.
- \( S_4 = (7, 8, 9, 10) \) has 4 numbers.
- \( S_5 = (11, 12, 13, 14, 15) \) has 5 numbers.

From this, we can see that \( S_n \) contains \( n \) numbers.

### Step 2: Determine the starting and ending numbers of \( S_n \)
To find the starting number of \( S_n \), we can calculate the total number of integers in all previous sets:
- The total number of integers in \( S_1, S_2, \ldots, S_{n-1} \) is given by the sum of the first \( n-1 \) natural numbers:
\[ \text{Total numbers} = 1 + 2 + 3 + \ldots + (n-1) = \frac{(n-1)n}{2} \]
Thus, the first number in \( S_n \) is:
\[ \text{First number in } S_n = \frac{(n-1)n}{2} + 1 \]
The last number in \( S_n \) will be:
\[ \text{Last number in } S_n = \frac{(n-1)n}{2} + n = \frac{n(n+1)}{2} \]

### Step 3: Find the first and last numbers in \( S_{50} \)
For \( n = 50 \):
- First number in \( S_{50} \):
\[ \text{First number} = \frac{(50-1) \cdot 50}{2} + 1 = \frac{49 \cdot 50}{2} + 1 = 1225 + 1 = 1226 \]
- Last number in \( S_{50} \):
\[ \text{Last number} = \frac{50 \cdot 51}{2} = 1275 \]

### Step 4: Calculate the sum of numbers from 1226 to 1275
The sum of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
where:
- \( n \) = number of terms,
- \( a \) = first term,
- \( l \) = last term.

#### Step 4.1: Find the number of terms
The number of terms from 1226 to 1275 is:
\[ n = 1275 - 1226 + 1 = 50 \]

#### Step 4.2: Calculate the sum
Using the sum formula:
\[ S_{50} = \frac{50}{2} \cdot (1226 + 1275) = 25 \cdot 2501 = 62525 \]

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

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