Question

The sum of squares of three consecutive natural number is 110. Determine the numbers.

Answer 1

${x}^{2} + {(x + 1)}^{2} + {(x + 2)}^{2} = 110$
$3 {x}^{2} + 6x + 5 = 110 \\ 3 {x}^{2} + 6x - 105 = 0 \\ 3 {x}^{2} + 21x - 15x - 105 = 0 \\ 3x(x - 7) - 15(x - 7) = 0 \\ x = 7 \: or - 5$

Answer 2

Let three consecutive natural numbers be x, x + 1 and x +2.

Then according to problem

(x)2 + (x + 1)2 + (x + 2)2 = 110

⇒ x2 + x2 + 1 + 2x + x2 + 4 + 4x – 110 = 0

⇒  3x2 + 6x – 105 = 0

⇒ x2 + 2x – 35 = 0

⇒ x2 + 7x – 5x – 35 = 0

⇒ x(x + 7) – 5(x + 7) = 0

⇒  (x + 7)(x – 5) = 0

⇒ x + 7 = 0 or x – 5 = 0

⇒ x  = -7 or  x = 5

But x = - 7 is rejected as it is not a natural number.

Then x = 5

Hence,  required numbers are 5, (5 + 1), (5 + 2)

i.e.,  5, 6  and 7.