The average of squares of five consecutive odd natural numbers is 233. What is the average of the largest number and the smallest number?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. 15.
Let the five consecutive natural numbers be (2k - 5), (2k - 3)(2k - 1)(2k + 1)(2k + 3).
Sum of squares of these five consecutive odd natural number,
⇒(2k - 5)2 + (2k - 3)2 + (2k - 1)2 + (2k + 1)2 + (2k + 3)2 = 233 × 5
⇒ 4k2 + 25 - 20k + 4k2 + 9 - 12k + 4k2 + 1 - 4k + 4k2 + 1 + 4k + 4k2 + 9 + 12k = 233 × 5 = 1165.
⇒20k2 - 20k - 1120 = 0
⇒ K2 - k - 56 = 0
⇒ K2 - 8k + 7k - 56 = 0
⇒ k(k - 8) + 7(k - 8) = 0
⇒ (k - 8)(k + 7) = 0
⇒ K = -7, 8.
Smallest number = 2k - 5 = 2 × 8 - 5 = 11
Largest number = 2k + 3 = 2 × 8 + 3 = 19.
Average of smallest and largest number = (11 + 19)/2 = 15.