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The sum of the squares of three consecutive natural numbers is 149. Find the numbers.

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Let the three consecutive natural numbers be ‘a’, ‘a + 1’ and ‘a + 2’ 

⇒ a2 + (a + 1)2 + (a + 2)2 = 149 

⇒ a2 + a2 + 2a + 1 + a2 + 4a + 4 = 149 

⇒ 3a2 + 6a - 144 = 0 

⇒ a2 + 2a – 48 = 0 

⇒ a2 + 8a – 6a – 48 = 0 

⇒ a(a + 8) -6(a + 8) = 0 

⇒ (a – 6)(a + 8) = 0 

⇒ a = 6 or a = -8, 

however a = -8 is not possible as -8 is not a natural number 

Numbers are 6, 7, 8

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