The sequence of sets `S_1, S_2,S_3,S_4,…..` is defined as `S_1 = {1}. S_2 = {3,5}, S_3 = {7,9,11}, S_4 = {13,15,17,19}`….etc.
The sum of all the elements of `S_(101)` :

Options

A. 1531441
B. 1189811
C. 1030301
D. none of these

Answer

To find the sum of all elements in the set \( S_{101} \), we will follow these steps:

### Step 1: Identify the first element of the set \( S_n \)

The first element of each set \( S_n \) can be expressed using the formula:
\[ \text{First element of } S_n = n^2 - n + 1 \]

For \( n = 101 \):
\[ \text{First element of } S_{101} = 101^2 - 101 + 1 \]
Calculating \( 101^2 \):
\[ 101^2 = 10201 \]
Now substituting back:
\[ \text{First element of } S_{101} = 10201 - 101 + 1 = 10201 - 100 = 10101 \]

### Step 2: Determine the number of elements in \( S_{101} \)

The number of elements in each set \( S_n \) is equal to \( n \). Therefore, the number of elements in \( S_{101} \) is:
\[ \text{Number of elements} = 101 \]

### Step 3: Identify the last element of the set \( S_{101} \)

The elements of \( S_{101} \) form an arithmetic sequence where:
- The first term \( a = 10101 \)
- The common difference \( d = 2 \)

The last term \( T_n \) of an arithmetic sequence can be calculated using the formula:
\[ T_n = a + (n - 1) \cdot d \]
Substituting the values:
\[ T_{101} = 10101 + (101 - 1) \cdot 2 \]
Calculating:
\[ T_{101} = 10101 + 100 \cdot 2 = 10101 + 200 = 10301 \]

### Step 4: Calculate the sum of all elements in \( S_{101} \)

The sum \( S_n \) of the first \( n \) terms of an arithmetic sequence can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + T_n) \]
Substituting the values:
\[ S_{101} = \frac{101}{2} \cdot (10101 + 10301) \]
Calculating the sum inside the parentheses:
\[ 10101 + 10301 = 20402 \]
Now substituting back:
\[ S_{101} = \frac{101}{2} \cdot 20402 = 101 \cdot 10201 \]
Calculating:
\[ S_{101} = 1030301 \]

### Final Answer:
The sum of all elements of \( S_{101} \) is \( 1030301 \).

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