To find the average of all even numbers between 104 and 148, we can follow these steps:
### Step 1: Identify the even numbers in the range
The first even number greater than 104 is 106, and the last even number less than 148 is 146. The even numbers between 104 and 148 are:
106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146.
### Step 2: Count the number of even numbers
To find the total number of even numbers, we can use the formula for the nth term of an arithmetic progression (AP):
- First term (a) = 106
- Last term (l) = 146
- Common difference (d) = 2
The formula for the nth term of an AP is given by:
\[ l = a + (n-1) \cdot d \]
Rearranging this gives:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{146 - 106}{2} + 1 = \frac{40}{2} + 1 = 20 + 1 = 21 \]
So, there are 21 even numbers.
### Step 3: Calculate the sum of the even numbers
The sum of an arithmetic series can be calculated using the formula:
\[ \text{Sum} = \frac{n}{2} \cdot (a + l) \]
Substituting the values we found:
\[ \text{Sum} = \frac{21}{2} \cdot (106 + 146) \]
Calculating the sum:
\[ \text{Sum} = \frac{21}{2} \cdot 252 = 21 \cdot 126 = 2646 \]
### Step 4: Calculate the average
The average is given by:
\[ \text{Average} = \frac{\text{Sum}}{n} \]
Substituting the values:
\[ \text{Average} = \frac{2646}{21} = 126 \]
### Final Answer
The average of all even numbers between 104 and 148 is **126**.
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