A={2,3,5,7,11,...89,97}
B={4,6,8,10,12,...98,100}
C={1,9,15,21,25,27,33...95,99}
The average of all the elements of B is :

Options

A. 52
B. 48
C. 49
D. none of these

Answer

To find the average of all the elements in set B, we will follow these steps:

### Step 1: Identify the elements in set B
Set B consists of even numbers starting from 4 up to 100. The elements are:
\[ B = \{4, 6, 8, 10, 12, \ldots, 98, 100\} \]

### Step 2: Determine the first term (a) and the common difference (d)
The first term \( a \) of set B is 4, and the common difference \( d \) is 2.

### Step 3: Find the last term (l)
The last term \( l \) of set B is 100.

### Step 4: Calculate the number of terms (n)
To find the number of terms in set B, we can use the formula for the nth term of an arithmetic progression:
\[ l = a + (n-1) \cdot d \]
Substituting the known values:
\[ 100 = 4 + (n-1) \cdot 2 \]
\[ 100 - 4 = (n-1) \cdot 2 \]
\[ 96 = (n-1) \cdot 2 \]
\[ n-1 = \frac{96}{2} \]
\[ n-1 = 48 \]
\[ n = 48 + 1 = 49 \]

### Step 5: Calculate the sum of the elements in set B
The sum \( S \) of an arithmetic progression can be calculated using the formula:
\[ S = \frac{n}{2} \cdot (a + l) \]
Substituting the values we found:
\[ S = \frac{49}{2} \cdot (4 + 100) \]
\[ S = \frac{49}{2} \cdot 104 \]
\[ S = 49 \cdot 52 \]
\[ S = 2548 \]

### Step 6: Calculate the average of the elements in set B
The average \( \text{Average} \) is given by:
\[ \text{Average} = \frac{S}{n} \]
Substituting the values:
\[ \text{Average} = \frac{2548}{49} \]
\[ \text{Average} = 52 \]

### Final Answer
The average of all the elements of set B is **52**.

---