The average of squares of consecutive odd numbers from 1 to 13 is

Options

A. 70
B. 65
C. 75
D. 66

Answer

To find the average of the squares of consecutive odd numbers from 1 to 13, we can follow these steps:

### Step 1: Identify the odd numbers from 1 to 13
The consecutive odd numbers from 1 to 13 are:
1, 3, 5, 7, 9, 11, 13

### Step 2: Calculate the squares of these odd numbers
Now, we will calculate the square of each of these numbers:
- \(1^2 = 1\)
- \(3^2 = 9\)
- \(5^2 = 25\)
- \(7^2 = 49\)
- \(9^2 = 81\)
- \(11^2 = 121\)
- \(13^2 = 169\)

So, the squares are:
1, 9, 25, 49, 81, 121, 169

### Step 3: Sum the squares
Next, we will sum these squares:
\[ 1 + 9 + 25 + 49 + 81 + 121 + 169 \]
Calculating the sum step-by-step:
- \(1 + 9 = 10\)
- \(10 + 25 = 35\)
- \(35 + 49 = 84\)
- \(84 + 81 = 165\)
- \(165 + 121 = 286\)
- \(286 + 169 = 455\)

So, the total sum of the squares is 455.

### Step 4: Count the number of observations
We have a total of 7 odd numbers (1, 3, 5, 7, 9, 11, 13).

### Step 5: Calculate the average
The average is calculated by dividing the sum of the squares by the total number of observations:
\[ \text{Average} = \frac{\text{Sum of squares}}{\text{Number of observations}} = \frac{455}{7} \]

Calculating this gives:
\[ \frac{455}{7} = 65 \]

### Conclusion
The average of the squares of consecutive odd numbers from 1 to 13 is **65**.

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