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

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3, 2, 5

3, 6, 2

6, 7, 8

2, 6, 6

answer is C.

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Detailed Solution

Concept: Take the first number to be x, therefore, the second and third number be (x+1) and (x+2) .
Here,

three consecutive number = x, x + 1 and x + 2

So,
According to question,

x^{2} + {(x + 1)}^{2} + {(x + 2)}^{2} = 149

x^{2} + x^{2} + 2 x + 1 + x^{2} + 4 x + 4 = 149

3 x^{2} + 6 x = 149 - 5

3 x^{2} + 6 x = 144

3 (x^{2} + 2 x) = 144

x^{2} + 2 x = 48

x² + 2 x - 48 = 0

x^{2} + 8 x - 6 x - 48 = 0

x (x + 8) - 6 (x + 8) = 0

(x + 8) (x - 6) = 0

Then,

if x + 8 = 0, x = - 8

if x - 6 = 0, x = 6

Given three consecutive numbers are natural numbers
Therefore,

cannot be - 8

So,

x, x + 1, x + 2 are 6, 6 + 1, 6 + 2

= 6, 7, 8

Hence, the correct answer is option (3) .

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