What will be the average of even numbers between 21 to 163?

Options

A. 87.5
B. 94
C. 96.5
D. 92

Answer

To find the average of even numbers between 21 and 163, we can follow these steps:

### Step 1: Identify the range of even numbers
The smallest even number greater than 21 is 22, and the largest even number less than or equal to 163 is 162. Therefore, the even numbers we are considering are from 22 to 162.

### Step 2: List the even numbers
The even numbers in this range are:
22, 24, 26, ..., 160, 162.

### Step 3: Recognize the sequence
These numbers form an arithmetic progression (AP) where:
- First term (a) = 22
- Last term (l) = 162
- Common difference (d) = 2

### Step 4: Find the number of terms (n)
To find the number of terms in the AP, we use the formula for the nth term of an AP:
\[ T_n = a + (n - 1) \cdot d \]
Setting \( T_n = 162 \):
\[ 162 = 22 + (n - 1) \cdot 2 \]
Subtract 22 from both sides:
\[ 140 = (n - 1) \cdot 2 \]
Divide by 2:
\[ 70 = n - 1 \]
Add 1 to both sides:
\[ n = 71 \]

### Step 5: Calculate the average
The average of an arithmetic progression can be calculated using the formula:
\[ \text{Average} = \frac{a + l}{2} \]
Substituting the values:
\[ \text{Average} = \frac{22 + 162}{2} \]
\[ \text{Average} = \frac{184}{2} \]
\[ \text{Average} = 92 \]

### Conclusion
The average of even numbers between 21 and 163 is **92**.

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