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 a least and a greatest element of Set C are transferred from Set C to Set A then the average of Set A :

Options

A. increases
B. decreases
C. remains constant
D. can't be determind

Answer

To solve the problem step by step, we will first identify the elements of Set C, then transfer the least and greatest elements to Set A, and finally calculate the new average of Set A.

### Step 1: Identify the elements of Set C
Set C is given as {3, 9, 15, 21, 27, 33, ..., 99}. This is an arithmetic sequence where the first term \( a = 3 \) and the common difference \( d = 6 \).

To find the number of terms in Set C, we can use the formula for the \( n \)-th term of an arithmetic sequence:
\[ a_n = a + (n-1) \cdot d \]
Setting \( a_n = 99 \):
\[ 99 = 3 + (n-1) \cdot 6 \]
\[ 96 = (n-1) \cdot 6 \]
\[ n-1 = 16 \]
\[ n = 17 \]

So, there are 17 elements in Set C.

### Step 2: Identify the least and greatest elements of Set C
The least element of Set C is 3, and the greatest element is 99.

### Step 3: Transfer the least and greatest elements from Set C to Set A
Set A initially contains prime numbers: {2, 3, 5, 7, 11, ..., 89, 97}. We will transfer 3 and 99 to Set A. However, since 3 is already in Set A, we only add 99.

Now, Set A becomes:
\[ A = \{2, 3, 5, 7, 11, ..., 89, 97, 99\} \]

### Step 4: Calculate the sum of the elements in Set A
To find the average, we need the sum of the elements in Set A. The sum of the prime numbers from 2 to 97 can be calculated. The prime numbers are:
\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 \]
The sum of these primes is 1060.

Now, we add 99 to this sum:
\[ \text{New Sum} = 1060 + 99 = 1159 \]

### Step 5: Calculate the number of elements in Set A
Initially, there were 25 prime numbers in Set A. After adding 99, the total number of elements in Set A becomes:
\[ \text{New Count} = 25 + 1 = 26 \]

### Step 6: Calculate the new average of Set A
The average is calculated as:
\[ \text{Average} = \frac{\text{Sum of elements}}{\text{Number of elements}} = \frac{1159}{26} \]

Calculating this gives:
\[ \text{Average} \approx 44.5769 \]

### Final Answer
The average of Set A after transferring the least and greatest elements from Set C is approximately 44.58.

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