Find the average of the squares of the consecutive odd numbers from 1 to 21.

CONCEPT

The average of a finite list is its total divided by the number of terms.

For the first n positive odd numbers, the sum of their squares is n(4n2 − 1)/3.

APPLICATION

1. The odd numbers from 1 to 21 are 1, 3, …, 21. Their count is n = (21 + 1)/2 = 11.

2. Apply the odd-square formula: 11(4 × 112 − 1)/3 = 11(4 × 121 − 1)/3 = 11(484 − 1)/3 = 11 × (483/3) = 11 × 161 = 1771.

3. Divide the total by the 11 terms: 1771/11 = 161.

CROSS-CHECK

1. Group the directly calculated squares from opposite ends: (12 + 212) + (32 + 192) + (52 + 172) + (72 + 152) + (92 + 132) + 112.

2. The grouped sum is 442 + 370 + 314 + 274 + 250 + 121 = 1771.

3. Dividing this independently checked total by 11 gives 1771/11 = 161. Therefore, the required average is 161.