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Question
The sum of the squares of three consecutive natural numbers is 110. Determine the numbers.
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Solution
Let three consecutive natural numbers be x, x + 1 and x + 2.
Then according to problem
(x)2 = (x + 1)2 + (x + 2)2 = 110
⇒ x2 + x2 + 1 + 2x + x2 + 4 + 4x - 110 = 0
⇒ 3x2 + 6x - 105 = 0
⇒ x2 + 2x - 35 = 0
⇒ x2 + 7x - 5x - 35 = 0
⇒ x(x + 7) - 5(x + 7) = 0
⇒ (x + 7) (x - 5) = 0
⇒ x + 7 = 0 or x - 5 = 0
⇒ x = -7 or x = 5
But x = -7 is rejected as it is not a natural number.
Then x = 5
Hence, required numbers are 5, (5 + 1), (5 + 2) i.e., 5, 6 and 7.
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