A={2,3,5,7,11,...89,97}
B={4,6,8,10,12,...98,100}
C={3,9,15,21,27,33...,99}
If an element greater than 50 of Set C is also included in the Set B,then the average of B :

Options

A. decreases by 3
B. increases by 2
C. decreases by 1
D. can't be determind

Answer

To solve the problem, we need to find the average of Set B and determine how it changes when we include an element greater than 50 from Set C.

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

### Step 2: Count the number of elements in Set B
The first term \( a = 4 \) and the last term \( l = 100 \). The common difference \( d = 2 \).

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

### Step 3: Calculate the sum of elements in Set B
The sum \( S \) of an arithmetic series 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 4: Calculate the average of Set B
The average \( \text{Average} \) of Set B is given by:
\[ \text{Average} = \frac{S}{n} \]
Substituting the values:
\[ \text{Average} = \frac{2548}{49} \]
\[ \text{Average} = 52 \]

### Step 5: Determine the effect of adding an element from Set C
Set C contains odd numbers starting from 3 up to 99. The elements greater than 50 in Set C are:
\[ C = \{51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99\} \]

If we add any element from Set C that is greater than 50 to Set B, we need to analyze how it affects the average.

- If we add 51 (the smallest element greater than 50), the new sum becomes \( 2548 + 51 = 2599 \) and the number of elements becomes \( 49 + 1 = 50 \).
- The new average would be:
\[ \text{New Average} = \frac{2599}{50} = 51.98 \]
This is less than 52, so the average decreases.

- If we add any other element greater than 51 (like 53, 55, etc.), the new average will be greater than 52.

### Conclusion
The average of Set B is 52. Including an element greater than 50 from Set C will decrease the average only when 51 is added; otherwise, it will increase.