Question : If the square of the sum of three positive consecutive natural numbers exceeds the sum of their squares by 292, what is the largest of the three numbers?
Option 1: 5
Option 2: 6
Option 3: 3
Option 4: 8
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Correct Answer: 8
Solution : The consecutive natural numbers are a = (x – 1), b = x, and c = (x + 1). According to the question, (a + b + c) 2 – (a 2 + b 2 + c 2 ) = 292 ⇒ a 2 + b 2 + c 2 + 2ab + 2bc + 2ca – a 2 – b 2 – c 2 = 292 ⇒ ab + bc + ca = 146 ⇒ (x)(x – 1) + (x)(x + 1) + (x + 1)(x – 1) = 146 ⇒ x 2 – x + x 2 + x + x 2 – 1 = 146 ⇒ 3x 2 = 147 ⇒ x 2 = $\frac{147}{3}$ ⇒ x 2 = 49 ⇒ x = 7 So, the largest number is (x + 1) = 8 Hence, the correct answer is 8.
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