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

Question

Question

This question was previously asked in

UGC NET Paper 1: Held on 8th Oct 2022 Shift 1

Answer 1

Given:

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

Concept used:

Odd number:

Odd numbers are the numbers that cannot be divided by 2 evenly.

For example: 1, 3, 5, 7 etc.

Consecutive numbers:

Numbers that follow each other continuously in the order from smallest to largest are called consecutive numbers. For example: 1, 2, 3, 4, 5, 6 etc.

Calculation:

The consecutive odd numbers from 1 to 21 are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21

To find the average of the squares of these numbers :

(1² + 3²+ 5² + 7² + 9² + 11² + 13² + 15² + 17² + 19² + 21²) / 11

⇒ (1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 + 289 + 361 + 441) / 11

⇒ 1771 / 11

⇒ 161

Therefore, the average of the squares of the consecutive odd numbers from 1 to 21 is approximately 161.

Shortcut Trick

The Formula for Sum of squares of first n odd numbers: [n(2n+1)(2n-1)] / 3
Here we have first 11 odd numbers; so average will be [11*23*21/3] / 11 = 161