Find the average of all the odd numbers and all the even numbers from 1 to 45.

Answer

To find the average of all the odd numbers and all the even numbers from 1 to 45, we will follow these steps:

### Step 1: Identify the odd numbers from 1 to 45.
The odd numbers in this range are:
1, 3, 5, 7, ..., 45.

### Step 2: Determine the number of odd terms (n).
The sequence of odd numbers can be expressed as:
- First term (A) = 1
- Last term (L) = 45
- Common difference (d) = 2

Using the formula for the nth term of an arithmetic sequence:
\[ T_n = A + (n - 1) \cdot d \]
Setting \( T_n = 45 \):
\[ 45 = 1 + (n - 1) \cdot 2 \]
\[ 45 - 1 = (n - 1) \cdot 2 \]
\[ 44 = (n - 1) \cdot 2 \]
\[ n - 1 = 22 \]
\[ n = 23 \]

### Step 3: Calculate the sum of the odd numbers.
Using the formula for the sum of an arithmetic series:
\[ S_n = \frac{n}{2} \cdot (A + L) \]
Substituting the values:
\[ S_{23} = \frac{23}{2} \cdot (1 + 45) \]
\[ S_{23} = \frac{23}{2} \cdot 46 \]
\[ S_{23} = 23 \cdot 23 = 529 \]

### Step 4: Calculate the average of the odd numbers.
The average is given by:
\[ \text{Average} = \frac{\text{Sum of odd numbers}}{\text{Number of odd numbers}} \]
\[ \text{Average} = \frac{529}{23} = 23 \]

### Step 5: Identify the even numbers from 1 to 45.
The even numbers in this range are:
2, 4, 6, 8, ..., 44.

### Step 6: Determine the number of even terms (n).
The sequence of even numbers can be expressed as:
- First term (A) = 2
- Last term (L) = 44
- Common difference (d) = 2

Using the formula for the nth term of an arithmetic sequence:
\[ T_n = A + (n - 1) \cdot d \]
Setting \( T_n = 44 \):
\[ 44 = 2 + (n - 1) \cdot 2 \]
\[ 44 - 2 = (n - 1) \cdot 2 \]
\[ 42 = (n - 1) \cdot 2 \]
\[ n - 1 = 21 \]
\[ n = 22 \]

### Step 7: Calculate the sum of the even numbers.
Using the formula for the sum of an arithmetic series:
\[ S_n = \frac{n}{2} \cdot (A + L) \]
Substituting the values:
\[ S_{22} = \frac{22}{2} \cdot (2 + 44) \]
\[ S_{22} = 11 \cdot 46 = 506 \]

### Step 8: Calculate the average of the even numbers.
The average is given by:
\[ \text{Average} = \frac{\text{Sum of even numbers}}{\text{Number of even numbers}} \]
\[ \text{Average} = \frac{506}{22} = 23 \]

### Final Result:
The average of all the odd numbers from 1 to 45 is 23, and the average of all the even numbers from 1 to 45 is also 23.

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