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Question
The sum of the squares of two consecutive odd natural numbers is 650. Find the two numbers.
Solution
Let the two consecutive od number be 2n + 1 and 2n + 3, where n is a natural number.
Given, (2n + 3)2 + (2n + 1)2 = 650
⇒ (4n2 + 9 + 12n) + (4n2 + 1 + 4n) = 650
⇒ 8n2 + 10 + 16n = 650
⇒ 8n2 + 16n - 640 = 0
⇒ n2 + 2n - 80 = 0
⇒ n2 + 10n - 8n - 80 = 0
⇒ n(n + 10) - 8(n + 10) = 0
⇒ (n - 8) (n + 10) = 0
Thus, n = 8 as n = -10 is not possible.
Thus, the two consecutive od natural numbers are 17 and 19.
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