The average of the squares of five consecutive odd natural numbers is 89. What is the average of the smallest and the largest of these numbers?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 9
Write the five numbers around the middle number \(n\): \(n-4,\ n-2,\ n,\ n+2,\ n+4\). The average of the smallest and the largest is then \(n\) itself.
Sum of the squares \(=5n^{2}+2\,(2^{2}+4^{2})=5n^{2}+40\), because the terms \(-4n\) and \(+4n\), and \(-8n\) and \(+8n\), cancel
Average of the squares \(=n^{2}+8=89\)
\(n^{2}=81\), and \(n=9\) because the numbers are natural numbers
The numbers are 5, 7, 9, 11, 13; average of 5 and 13 \(=9\)
The average of the smallest and the largest number is 9.