The average of the square of five consecutive odd natural number is 8…

Take the square root of 89 and approximately this will give the middle term.

\(\sqrt {89}\) = 9 approx.

1st    2nd    3rd    4th    5th

 5       7        9      11     13

So, Required average = \(\frac{13+5}{2}\) = 9

 

Alternate Solution:

Let the consecutive odd numbers are = (a - 4), (a - 2), a, (a + 2), (a + 4)

Sum of square of numbers =  (a - 4)² + (a - 2)² + a² + (a + 2)² + (a + 4)²

= (a² + 16 - 8a) + (a² + 4 - 4a) + a² + (a² + 4 + 4a) + (a² + 16 + 8a)

= 5a² + 40

Average of square of numbers = \(\frac{sum\;of\;square\;of\;numbers}{total\;numbers}\)

ATQ,

Average of square of numbers ⇒ \(\frac{5a^2\;+\;40}{5}\) =  89

⇒ a²+ 8 =  89

⇒ a² = 81

⇒ a = 9

So,

The largest number =  9 + 4 = 13

The smallest number = 9 - 4 = 5

Required average = \(\frac{13+5}{2}\) = 9