The sum of the squares of three consecutive natural numbers is 194. What is the sum of the numbers ?

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Q. The sum of the squares of three consecutive natural numbers is 194. What is the sum of the numbers ?

Options:
(a) 24 (b) 27 (c) 21 (d) 30

Solution:

Let the three consecutive natural numbers be: x, (x+1), (x+2)

According to the question:

x² + (x+1)² + (x+2)² = 194

⇒ x² + x² + 2x + 1 + x² + 4x + 4 = 194   
⇒ 3x² + 6x + 5 = 194
⇒ 3x²+6x = 189
⇒ x²+2x−63 = 0

⇒ (x+9)(x−7) = 0
⇒ x = 7

Numbers = 7, 8, 9
Sum = 7 + 8 + 9 = 24

Ans: 24