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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