What is the average of squares of consecutive odd numbers between 1 to 13 ?

Answer

To find the average of the squares of consecutive odd numbers between 1 and 13, we will follow these steps:

### Step 1: Identify the consecutive odd numbers between 1 and 13.
The odd numbers in this range are:
1, 3, 5, 7, 9, 11, 13

However, since we are looking for odd numbers **between** 1 and 13, we will exclude 1 and 13. Thus, the consecutive odd numbers are:
3, 5, 7, 9, 11

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

### Step 3: Sum the squares.
Next, we will add these squares together:
\[ 9 + 25 + 49 + 81 + 121 \]

Calculating this step-by-step:
- \(9 + 25 = 34\)
- \(34 + 49 = 83\)
- \(83 + 81 = 164\)
- \(164 + 121 = 285\)

So, the sum of the squares is \(285\).

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

### Step 5: Calculate the average.
To find the average, we divide the sum of the squares by the number of odd numbers:
\[ \text{Average} = \frac{\text{Sum of squares}}{\text{Number of odd numbers}} = \frac{285}{5} \]

Calculating this gives:
\[ 285 \div 5 = 57 \]

### Final Answer:
The average of the squares of consecutive odd numbers between 1 and 13 is **57**.

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