There are 3 sets of natural numbers 1 ot 100. Set A contains all the natural numbers which are prime, upto 100. Set B contains all the non-prime even natural numbers upto 100. Set C contains all the non-prime odd natural numbers upto 100 i.e.,
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 of Set B
Set B contains all the non-prime even natural numbers up to 100. The elements of Set B are:
\[ B = \{4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100\} \]

### Step 2: Count the number of terms in Set B
To find the number of terms (n) in Set B, we can use the formula for the nth term of an arithmetic sequence:
\[ l = a + (n - 1) \cdot d \]
Where:
- \( l \) is the last term (100),
- \( a \) is the first term (4),
- \( d \) is the common difference (2).

Rearranging the formula gives:
\[ 100 = 4 + (n - 1) \cdot 2 \]

### Step 3: Solve for n
Subtract 4 from both sides:
\[ 96 = (n - 1) \cdot 2 \]
Now divide both sides by 2:
\[ 48 = n - 1 \]
Adding 1 to both sides gives:
\[ n = 49 \]

### Step 4: Calculate the sum of the elements in Set B
The sum of the first n terms of an arithmetic sequence can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we have:
\[ S_{49} = \frac{49}{2} \cdot (4 + 100) \]
\[ S_{49} = \frac{49}{2} \cdot 104 \]
Calculating \( \frac{104}{2} \):
\[ S_{49} = 49 \cdot 52 \]

### Step 5: Calculate the average of Set B
The average can be calculated using the formula:
\[ \text{Average} = \frac{\text{Sum of terms}}{n} \]
Substituting the values we have:
\[ \text{Average} = \frac{49 \cdot 52}{49} \]
The \( 49 \) cancels out:
\[ \text{Average} = 52 \]

### Final Answer
The average of all the elements of Set B is:
\[ \text{Average} = 52 \]

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